Introduction

Quantum Information Processing

Superconducting Qubits and Circuit QED

Circuit QED: Quantum Information Processing with a Photon Bus

Experimental Setup and Details

Initialization and Benchmarking of Single-Qubit Gates

Two-Qubit Circuit QED: Riding the Quantum Bus

Entanglement On-Demand and Joint Readout

Two-Qubit Algorithms

Conclusions and Future Work

Bibliography
© 2010 by Jerry Moy Chow
All rights reserved.
Quantum Information Processing with Superconducting Qubits

A Dissertation
Presented to the Faculty of the Graduate School
of
Yale University
in Candidacy for the Degree of
Doctor of Philosophy

by
Jerry Moy Chow

Dissertation Director: Professor Robert J. Schoelkopf

May 2010
 Abstract
Quantum Information Processing with Superconducting Qubits
Jerry Moy Chow
2010

This thesis describes the theoretical framework, implementation, and measurements of a

quantum processor comprised of superconducting qubits coupled in the circuit quantum
electrodynamics (QED) architecture. In the realization of circuit QED, two superconducting
‘transmon’ charge qubits are capacitively coupled to a one-dimensional microwave transmis-
sion line resonator which serves as a quantum bus. Single-qubit rotations can be applied
through the resonator and their operation is characterized using various benchmarking
techniques. Through a virtual photon interaction via the quantum bus, the two qubits can
coherently swap a single excitation. A separate two-qubit conditional phase interaction is
also observed which is attributable to an interaction in the two-excitation manifold of the
transmons. Furthermore, the same quantum bus which couples the qubits can be used as a
joint detector of the full two-qubit quantum state. Entanglement witnesses and a violation
of a Bell-type inequality are found using this joint detector on highly entangled states. Fi-
nally, combining the single-qubit rotations, conditional phase interaction, and joint readout,
allows the realization and characterization of simple quantum algorithms, specifically the
Deutsch-Jozsa and Grover’s search algorithms.
 Contents

Contents v

List of Figures xi

List of Tables xv

Acknowledgements xvii

Publication list xix

Nomenclature xxi

1 Introduction 1
1.1 Computing with quantum mechanics . . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Experimental implementations of quantum processors . . . . . . . . . . . . . 4
1.3 Overview of thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6

2 Quantum Information Processing 9

2.1 Universal quantum computing . . . . . . . . . . . . . . . . . . . . . . . . . . . 10
2.2 Single-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
2.3 Two-qubit entanglement gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.1 cNOT gate . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
2.3.2 c-Phase gate√ (cU i j ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
2.3.3 iSWAP and iSWAP gates . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.4 Quantum algorithms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4.1 Quantum parallelism in an algorithm . . . . . . . . . . . . . . . . . . 21
2.4.2 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 23

v
vi contents

2.4.3 Grover’s search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 25

2.4.4 Shor’s and other quantum algorithms . . . . . . . . . . . . . . . . . . . 29
2.5 Quantum measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.5.1 Density matrix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31
2.5.2 State tomography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Entanglement metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
2.6.1 Concurrence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36
2.6.2 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 Bell tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.7.1 Clauser-Horne-Shimony-Holt inequality . . . . . . . . . . . . . . . . 40
2.7.2 CHSH entanglement witness . . . . . . . . . . . . . . . . . . . . . . . . 43
2.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 43

3 Superconducting Qubits and Circuit QED 45

3.1 Superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
3.1.1 Josephson junction as a non-linear inductor . . . . . . . . . . . . . . . 46
3.1.2 The Cooper-pair box qubit . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.1.3 The transmon qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.2 Coupling superconducting qubits . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.1 Fixed capacitive coupling . . . . . . . . . . . . . . . . . . . . . . . . . . 53
3.2.2 Tunable inductive coupling . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.2.3 Quantum bus coupling . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.3 Cavity quantum electrodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.3.1 Strong coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.3.2 Dispersive coupling regime . . . . . . . . . . . . . . . . . . . . . . . . . 60
3.4 Circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 Coupling a transmon to a coplanar waveguide resonator . . . . . . . 61
3.4.2 Dispersive regime of circuit QED . . . . . . . . . . . . . . . . . . . . . 65
3.4.3 Strong dispersive regime . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5 Qubit decoherence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66
3.5.1 Relaxation and the Purcell effect . . . . . . . . . . . . . . . . . . . . . . 68
3.5.2 Dephasing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70
3.6 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

4 Circuit QED: Quantum Information Processing with a Photon Bus 75

4.1 Initialization . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76
4.2 Single-qubit gates in circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.1 Introducing a drive . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77
4.2.2 X-Y gates for a qubit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
4.2.3 X-Y gates for a transmon multi-level atom . . . . . . . . . . . . . . . . 80
4.2.4 Z (phase) gates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 85
4.3 Two-qubit gates in circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . 87
 contents vii

4.3.1 Two-qubits in the dispersive regime . . . . . . . . . . . . . . . . . . . 88

4.3.2 Virtual qubit-qubit interaction . . . . . . . . . . . . . . . . . . . . . . . 89
4.3.3 σz ⊗ σz higher level transmon interaction . . . . . . . . . . . . . . . . 92
4.4 Muliplexed joint qubit readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.4.1 Deriving the measurement operator . . . . . . . . . . . . . . . . . . . 97
4.4.2 State tomography in circuit QED . . . . . . . . . . . . . . . . . . . . . 101
4.4.3 Entanglement by joint measurement . . . . . . . . . . . . . . . . . . . 103
4.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104

5 Experimental Setup and Details 105

5.1 Experimental test samples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2 Resonator Fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.1 Resonator parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . 106
5.2.2 Optical lithography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Transmon fabrication . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
5.3.1 ‘Traditional’ transmon . . . . . . . . . . . . . . . . . . . . . . . . . . . 110
5.3.2 Balanced design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.3.3 Flux-bias transmon design . . . . . . . . . . . . . . . . . . . . . . . . . 112
5.4 Sample boards and holders . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
5.4.1 Coffin design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.4.2 Octobox design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121
5.5 Cryogenic setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.6 Room temperature control . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 127
5.7 Pulse control and modulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129
5.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 129

6 Initialization and Benchmarking of Single-Qubit Gates 131

6.1 Initializing pure states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132
6.1.1 Nonlinear vacuum Rabi . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
6.1.2 Cavity temperature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.1.3 Vacuum Rabi summary . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2 Characterizing single-qubit gates . . . . . . . . . . . . . . . . . . . . . . . . . . 141
6.3 Single-qubit gate error experiments . . . . . . . . . . . . . . . . . . . . . . . . 142
6.3.1 Microwave pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.3.2 Calibration of single-qubit gates . . . . . . . . . . . . . . . . . . . . . . 145
6.3.3 Single qubit readout calibration . . . . . . . . . . . . . . . . . . . . . . 145
6.3.4 Double π metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3.5 Quantum process tomography . . . . . . . . . . . . . . . . . . . . . . . 149
6.3.6 Randomized benchmarking . . . . . . . . . . . . . . . . . . . . . . . . 153
6.3.7 Summary of error metrics . . . . . . . . . . . . . . . . . . . . . . . . . 156
6.4 Derivative-based pulse shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
6.4.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159
viii contents

6.4.2 Results with standard pulse shaping . . . . . . . . . . . . . . . . . . . 160

6.4.3 Experimentally implementing derivative pulse shaping . . . . . . . . 162
6.4.4 Summary of DPS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 166
6.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

7 Two-Qubit Circuit QED: Riding the Quantum Bus 169

7.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
7.2 Two-qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
7.2.1 Qubit-qubit avoided crossing . . . . . . . . . . . . . . . . . . . . . . . 175
7.2.2 The dark state . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 175
7.3 Multiplexed joint qubit readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 178
7.4 Coherent state transfer: Stark swap . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.5 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 185

8 Entanglement On-Demand and Joint Readout 187

8.1 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189
8.2 Virtual swap interaction via flux bias . . . . . . . . . . . . . . . . . . . . . . . . 192
8.3 Higher-level transmon interaction . . . . . . . . . . . . . . . . . . . . . . . . . 194
8.3.1 Tuning up a c-Phase gate . . . . . . . . . . . . . . . . . . . . . . . . . . 196
8.3.2 Generating Bell states . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.4 Joint readout of two qubits . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 199
8.5 Calibrating the measurement model . . . . . . . . . . . . . . . . . . . . . . . . 201
8.6 Quantum state tomography and the Pauli set . . . . . . . . . . . . . . . . . . . 204
8.6.1 The density matrix representation . . . . . . . . . . . . . . . . . . . . . 205
8.6.2 Biasing of metrics by maximum-likelihood estimation . . . . . . . . 208
8.6.3 The Pauli set representation . . . . . . . . . . . . . . . . . . . . . . . . 209
8.7 Characterizing the quantum states . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.7.1 Fidelity to targeted states . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.7.2 Entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . . . . . 213
8.7.3 Clauser-Horne-Shimony-Holt inequality violation . . . . . . . . . . . 214
8.8 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216

9 Two-Qubit Algorithms 219

9.1 Experimental details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 220
9.2 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 221
9.2.1 Breaking down the algorithm . . . . . . . . . . . . . . . . . . . . . . . 221
9.2.2 Deutsch–Jozsa results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
9.3 Grover search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.3.1 The oracle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
9.3.2 Breaking down the algorithm . . . . . . . . . . . . . . . . . . . . . . . 226
9.3.3 Grover results and debugger . . . . . . . . . . . . . . . . . . . . . . . . 227
9.4 Chapter summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 231
 contents ix

10 Conclusions and Future Work 233

10.1 Improving one and two-qubit operations . . . . . . . . . . . . . . . . . . . . . 234
10.1.1 Longer coherence times . . . . . . . . . . . . . . . . . . . . . . . . . . . 234
10.1.2 Better qubit operations . . . . . . . . . . . . . . . . . . . . . . . . . . . 235
10.2 More qubits in circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 237
10.3 Quantum information outlook . . . . . . . . . . . . . . . . . . . . . . . . . . . 240

Bibliography 243

Appendices 258
A Mathematica code for microwave pulse generation 259

B Mathematica code for tomography 281

Copyright Permissions 297

 List of Figures

2.1 The Bloch sphere . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12

2.2 Circuit representation for the controlled NOT gate . . . . . . . . . . . . . . . 15
2.3 Circuit representation for the conditional phase gate . . . . . . . . . . . . . . 17
2.4 Circuit form for constructing a cNOT from √ c-Phase . . . . . . . . . . . . . . . 18
2.5 Circuit form for constructing a cNOT from iSWAP or iSWAP . . . . . . . 19
2.6 Anatomy of a quantum algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 22
2.7 Deutsch-Jozsa algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
2.8 Grover’s search algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.9 Cartoon state illustration of the Grover iteration . . . . . . . . . . . . . . . . . 28
2.10 Entanglement of formation versus concurrence . . . . . . . . . . . . . . . . . 37
2.11 State space and entanglement witnesses . . . . . . . . . . . . . . . . . . . . . . 39
2.12 Schematic for the CHSH test . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.1 The Cooper pair box . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Charge dispersion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50
3.3 Schemes for coupling charge qubits . . . . . . . . . . . . . . . . . . . . . . . . . 54
3.4 Charge-qubit coupling networks . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.5 Charge-qubit quantum bus . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
3.6 Illustration of Cavity QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.7 Illustration of circuit QED . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.8 Reduced transmon coupling to CPW network . . . . . . . . . . . . . . . . . . 63
3.9 Strong dispersive regime of circuit QED . . . . . . . . . . . . . . . . . . . . . . 67
3.10 Environmental coupling of a transmon . . . . . . . . . . . . . . . . . . . . . . 70
3.11 Flux noise contribution to dephasing . . . . . . . . . . . . . . . . . . . . . . . . 73

4.1 Frequency bandwidth of Gaussian pulse shapes . . . . . . . . . . . . . . . . . 83

4.2 Error per gate with and without DRAG . . . . . . . . . . . . . . . . . . . . . . 84

xi
xii list of figures

4.3 Virtual Swap . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

4.4 Two excitation manifold . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 93
4.5 Level scheme for c-Phase . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94
4.6 Transmission in strong dispersive regime for two qubits . . . . . . . . . . . . 98
4.7 Measurement model coefficients versus drive frequency . . . . . . . . . . . . 101
4.8 Dispersive peaks for generating Bell states by measurement . . . . . . . . . . 104

5.1 Coplanar waveguide geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

5.2 Optical images of resonator topologies . . . . . . . . . . . . . . . . . . . . . . . 108
5.3 Optical images of different transmon designs . . . . . . . . . . . . . . . . . . . 110
5.4 The capacitance network for the transmon in a coplanar waveguide resonator 111
5.5 The flux-bias line . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114
5.6 FBL schematic for Sonnet simulations . . . . . . . . . . . . . . . . . . . . . . . 115
5.7 Sonnet simulations of resonators with and without FBLs . . . . . . . . . . . . 116
5.8 Current density simulations of resonators with FBLs . . . . . . . . . . . . . . 117
5.9 Pushing up the wiggle-waggle with bondwires . . . . . . . . . . . . . . . . . . 118
5.10 Optical image of an on-chip wirebond . . . . . . . . . . . . . . . . . . . . . . . 119
5.11 Experimental transmission spectrum with and without wirebond . . . . . . . 120
5.12 Coffin box holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122
5.13 Octobox holder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123
5.14 Resonator with FBL in octobox . . . . . . . . . . . . . . . . . . . . . . . . . . . 124
5.15 Schematic of cryogenic circuitry . . . . . . . . . . . . . . . . . . . . . . . . . . 126
5.16 Room temperature control schematic . . . . . . . . . . . . . . . . . . . . . . . 128

6.1 Vacuum Rabi splitting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134

6.2 Jaynes-Cummings ladder . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134
6.3 √
Transmission versus magnetic field and drive frequency . . . . . . . . . . . . 136
6.4 Emergence of n peaks under strong driving of the vacuum Rabi transition 137
6.5 Strongly driven vacuum Rabi at elevated temperature . . . . . . . . . . . . . . 140
6.6 Microwave pulse shapes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 143
6.7 Sample sequence of concatenated pulses . . . . . . . . . . . . . . . . . . . . . . 144
6.8 Bang-bang gate characterization and visibility . . . . . . . . . . . . . . . . . . 147
6.9 Schematic for quantum process tomography . . . . . . . . . . . . . . . . . . . 149
6.10 Quantum process tomography experimental results . . . . . . . . . . . . . . . 151
6.11 Gate errors from QPT . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152
6.12 Pulse sequence schematic for randomized benchmarking . . . . . . . . . . . 155
6.13 Randomized benchmarking 3 ns pulse width . . . . . . . . . . . . . . . . . . . 156
6.14 Trace by trace randomized benchmarking . . . . . . . . . . . . . . . . . . . . . 157
6.15 Error per gate versus pulse width . . . . . . . . . . . . . . . . . . . . . . . . . . 158
6.16 Clifford averaged RB for cQED222 with standard pulse shaping . . . . . . . . 160
6.17 Error per gate with normal pulse shaping . . . . . . . . . . . . . . . . . . . . . 161
6.18 Composite x and y rotations with standard Gaussian pulse shaping . . . . . . 162
 list of figures xiii

6.19 Gaussian and derivative on I and Q quadratures . . . . . . . . . . . . . . . . . 163

6.20 Composite x and y rotations with derivative pulse shaping . . . . . . . . . . . 164
6.21 Randomized benchmarking for various pulse widths using derivative pulse
shaping . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
6.22 Trace by trace with and without derivative pulse shaping . . . . . . . . . . . . 167
6.23 Error per gate with derivative pulse shaping . . . . . . . . . . . . . . . . . . . . 168

7.1 Sample and scheme used to couple two qubits to an on-chip microwave cavity171
7.2 Strong coupling of two superconducting qubits . . . . . . . . . . . . . . . . . . 173
7.3 Two qubit dispersive cavity shifts . . . . . . . . . . . . . . . . . . . . . . . . . . 173
7.4 Scheme of the virtual photon swap interaction . . . . . . . . . . . . . . . . . . 174
7.5 Two qubit spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.6 Independent Rabi driving of two qubits . . . . . . . . . . . . . . . . . . . . . . 179
7.7 Two qubit multiplexed readout . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
7.8 Two qubit Stark shift spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . 181
7.9 Coherent swap protocol . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 182
7.10 Coherent state exchange . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
7.11 Stark swap frequency . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184

8.1 Schematic for two-qubit quantum bus with on-chip flux bias lines . . . . . . 190
8.2 Single excitation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
8.3 Flux bias swap experiments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
8.4 Two excitation spectroscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 195
8.5 Agreement of the splitting between experiment and theory . . . . . . . . . . . 197
8.6 Conditional phase gate tune-up sequences . . . . . . . . . . . . . . . . . . . . 198
8.7 Experimental protocols for generating Bell states . . . . . . . . . . . . . . . . . 200
8.8 Measurement transients for joint readout . . . . . . . . . . . . . . . . . . . . . 201
8.9 Rabi oscillations for readout characterization . . . . . . . . . . . . . . . . . . . 202
8.10 Fourier transforms of Rabi oscillations . . . . . . . . . . . . . . . . . . . . . . . 203
8.11 Density matrix representation of Bell states . . . . . . . . . . . . . . . . . . . . 206
8.12 Bias of entanglement metrics from MLE . . . . . . . . . . . . . . . . . . . . . . 209
8.13 Pauli set representation of two-qubit states . . . . . . . . . . . . . . . . . . . . 211
8.14 Pauli set for separable and entangled states differing only by a single-qubit
rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 212
8.15 Entanglement witness for separable states . . . . . . . . . . . . . . . . . . . . . 214
8.16 Entanglement witness for entangled states . . . . . . . . . . . . . . . . . . . . . 215
8.17 CHSH for separable states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
8.18 CHSH for entangled states . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217

9.1 Quantum circuit for DJ algorithm . . . . . . . . . . . . . . . . . . . . . . . . . 222

9.2 Results for four cases in DJ algorithm . . . . . . . . . . . . . . . . . . . . . . . 224
9.3 Two-qubit Grover algorithm schematic . . . . . . . . . . . . . . . . . . . . . . 226
xiv list of figures

9.4 Microwave and flux pulses for Grover algorithm . . . . . . . . . . . . . . . . . 228

9.5 Implementing Grover’s algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 230
9.6 Grover fidelity for all choice of oracles . . . . . . . . . . . . . . . . . . . . . . . 231

10.1 Four qubit sample . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 238

10.2 Three qubit protocols . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 239
 List of Tables

5.1 Summary of measured sample parameters . . . . . . . . . . . . . . . . . . . . 130

6.1 Gate errors for the three metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 158

8.1 The 30 raw measurements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

xv
for James
 Acknowledgements

F
irst I would like to thank my advisor Rob Schoelkopf for giving me the amazing
opportunity to work on superconducting quantum computing in RSL. From late nights
in the lab to phone calls while he is driving to work, he has taught me countless things to
become a better scientist. I’m very grateful for his patience for those times when I have
scurried onto some experimental tangent, and even more thankful for his unique quantum
engineer’s perspective.
I have always been amazed by how Steve Girvin could step into a complicated discussion,
get caught up within five minutes, and immediately begin to make experimental suggestions
which we might never have considered. Michel Devoret has shown me multiple times how
broad and how beautiful physics can be, through innocent experimental discussions that
blossom into philosophical treatises. Another very important individual and mainstay of RSL
has been Luigi Frunzio. Much thanks goes to him for the fabrication of the samples which I
have worked with. Furthermore, I have deeply appreciated his accessibility for discussions
and enthusiasm for new ideas and results.
Through my graduate career at Yale, I have also had the pleasure to work with a number
of truly remarkable experimental postdoctoral associates. The day-to-day interactions with
them, and team-bonding have helped shape my development as a scientist. First, there was
Johannes Majer, with whom I first plunged into the exciting experiments presented in this
thesis. I found it amazing that any given night in the lab with him could quickly transition
from taking critical experimental data to reveling and socializing at GPSCY. Andrew Houck
was ever the happy-go-lucky physicist, with a knack for always pointing me towards the most
efficient data set, so as to make time to go outside and fit in a round of wiffleball. Finally,
Leonardo DiCarlo has helped fill these final few years with some of the most productive

xvii
xviii acknowledgements

and ground-breaking science. He has shown me what true passion in science means and I
am very grateful for that.
On the theory side, the work and advice of postdocs Jay Gambetta and Jens Koch have
been immeasurable. From completing the J 4 team on the original cavity bus with myself and
Johannes, to going out for beers and working out at the gym, to always helping me understand
the gigabytes of data I’ve taken, they have rounded-out my scientific experience at Yale by
blurring the line between theorists and experimentalists.
The camaraderie and success of the fourth floor of Becton are truly remarkable. I especially
want to thank Lev Bishop, for always being there for me, from helping understand the
vacuum Rabi data of chapter 6, to being patient and waiting whenever I ask him to ‘hold on.’
I have never met anyone quite like the ‘idea-machine’ David Schuster, with whom I have
enjoyed playing basketball and discussing zany start-up projects. Then, there are also the
grad students with whom I entered the Yale physics department. Since day one, they have
always been accessible for hanging out and it has been absolutely wonderful to have shared
the grad school experience and developed life friendships with them.
Words cannot simply describe the bond and all the shared experiences between myself
and Blake Johnson while chugging through graduate school together. Blake has become
like a brother to me and his constant presence and cool demeanor have kept me grounded
through some of the most stressful periods. I also definitely have to thank him and his wife
Phyllis for providing me with so many excellent meals and truly helping me mature as a
person.
Finally, there are my friends and family back home, who have been a constant source of
support and love. I am very grateful to have been able to be so physically close to them, so
that I could visit on a whim and have some of my mother’s home cooked meals. I need to
especially thank my father, who’s confidence and teaching have guided me towards this
degree. From my brother James, to my aunts, and to all my cousins who check up on me, the
connection to family has been an absolute driving force of my desire to learn and to achieve.
I have to also thank my dear Charlotte for giving me perspective and connecting with my
heart at all times, apart and together.
 Publication list

This thesis is based in part on the following published articles:

1. J. Majer, J. M. Chow, J. M. Gambetta, J. Koch, B. R. Johnson, J. A. Schreier, L. Frunzio,

D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin, and
R. J. Schoelkopf, “Coupling superconducting qubits via a cavity bus,” Nature 449,
443–447 (2007).

2. J. M. Chow, J. M. Gambetta, L. Tornberg, J. Koch, L. S. Bishop, A. A. Houck, B. R.

Johnson, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Randomized benchmarking
and process tomography for gate errors in a solid-state qubit,” Phys. Rev. Lett. 102,
090502 (2009).

3. L. S. Bishop, J. M. Chow, J. Koch, A. A. Houck, M. H. Devoret, E. Thuneberg, S. M.

Girvin, and R. J. Schoelkopf, “Nonlinear response of the vacuum Rabi resonance,”
Nature Phys. 5, 105–109 (2009).

4. L. DiCarlo, J. M. Chow, J. M. Gambetta, L. S. Bishop, B. R. Johnson, D. I. Schuster,

J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, “Demonstration
of two-qubit algorithms with a superconducting quantum processor,” Nature 460,
240–244 (2009).

5. J. M. Chow, L. DiCarlo, J. M. Gambetta, A. Nunnenkamp, L. S. Bishop, L. Frunzio,

M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, “Entanglement metrology using a
joint readout of superconducting qubits,” arXiv:0908.1955 [cond-mat].

xix
 Nomenclature

Abbreviations:
CHSH Clauser-Horne-Shimony-Holt, see section 2.7.
CPB Cooper-pair box, see section 3.1.2.
c-Phase conditional-phase gate, see section 2.3.2.
cNOT controlled-NOT, see section 2.1.
DJ Deutsch-Jozsa, see section 2.4.2.
FBL flux-bias line, see section 5.3.3.
IC integrated circuit, see chapter 1.
JC Jaynes-Cummings, see section 3.3.
NMR nuclear magnetic resonance, see section 1.2.
PCB printed circuit board, see section 5.4.1.
POVM positive operator-valued measure, see section 2.5.
QED quantum electrodynamics, see section 1.3.
QFT quantum fourier transform, see section 2.4.4.
QIP quantum information processing, see section 1.3.
RF radio-frequency, see section 1.2.
RSA Rivest, Shamir, and Adleman, see section 1.1.
RWA rotating wave approximation, see section 3.3.

xxi
xxii nomenclature

SQUID superconducting quantum interference device, see section 3.1.2.

¹
Latin Letters:
B bound to concurrence, see section 2.6.2.
Cg qubit-cavity coupling capacitance, see (3.34).
C Clauser-Horne-Shimony-Holt operator, see section 2.7.
C concurrence, see section 2.6.1.
cU i j conditional-phase gate, conditioned on state i j, see section 2.3.2.
CΣ total capacitance to ground of charge qubit, see section 3.1.2.
D derivative pulse-shape amplitude scale factor, see section 6.4.3.
EC electrostatic charging energy, see section 3.1.2.
EJ Josephson energy, see section 3.1.2.
Em energy of m-th transmon level, see section 4.2.4.
F state fidelity, see (2.42).
G Grover iteration, see section 2.4.3.
i j transmon dipole coupling energy between charge levels i and j, see (3.36).
 vacuum Rabi coupling frequency, see (3.27).
H (i) Hadamard gate on qubit i, see section 2.2.
H ⊗n n-qubit simultaneous Hadamard gate, see section 2.4.1.
I single-qubit identity operator, also defined as 1, see section 2.2.
iSWAP i-swap gate, see section 2.3.3.
1 single-qubit identity operator, also defined as I, see section 2.2.
J virtual photon qubit-qubit swap interaction strength, see (4.34).
M measurement operator, see sections 2.5 and 4.4.1.
n̂ integer-valued Cooper pair number operator, see section 3.1.2.
n̄ mean number of photons in the cavity, also defined as ⟨n⟩, see section 3.4.3.
ng gate charge, see section 3.1.2.
 nomenclature xxiii

P1 qubit excited state population, see ??.

P state purity, see section 2.6.
RQ two-qubit Pauli operators, where R, Q ∈ I, X, Y , Z, and also referred to as R ⊗ Q,
see section 2.5.2.
Q quality factor of cavity, see section 5.2.
R i (θ) rotation around the axis i by angle θ, also defined as R θi , see (2.3).
√
iSWAP square-root of i-swap gate, see section 2.3.3.
T1 qubit relaxation time, see section 3.5.1.
Tϕ qubit dephasing time, see section 3.5.2.
T2 qubit decoherence time, see section 3.5.2.
V0 zero point root mean squared voltage in the cavity, see (3.34).
W entanglement witness, see section 2.6.2.
X, Y , Z single-qubit Pauli operator, also defined as σx,y,z , see (2.2).
z-cNOT zero controlled-NOT gate, see section 2.4.2.
¹
Greek Letters:
αm absolute anharmonicity of the m-th transmon level, see (3.18).
r
αm relative anharmonicity of the m-th transmon level, see (3.19).
β voltage division ratio, see (3.34).
βi sensitivity of the measurement to a specific Pauli operator indexed by i, see
section 4.4.1.
єm charge dispersion of the m-th transmon level, see (3.15).
χ state dependent cavity shift, see (3.45).
χ mn positive superoperator process determined by quantum process tomography, see
section 6.3.5.
Φ̃ external magnetic flux, see section 3.1.2.
ϕ̂ Josephson phase operator, see section 3.1.2.
Φ0 magnetic flux quantum, see section 3.1.2.
xxiv nomenclature

θz z-rotation phase of the qubit, see (4.30).

ρ density matrix, see section 2.5.
σx,y,z single-qubit Pauli operator, also defined as X, Y, Z, see (2.2).
ωC cavity excitation frequency, see (3.27).
ωk transmon transition frequency to level k, see (3.21).
ωq qubit transition frequency, see (3.20).
ωd qubit drive frequency, see section 7.2.2.
ζ two-qubit σz ⊗ σz interaction strength, see section 4.3.3.
 CHAPTER 1

Introduction

T
he ubiquity of computers and other devices with microprocessors reflects one of the more
successful technological developments over the past few decades. When the first solid-
state transistor was made in 1947 by John Bardeen, Walter Brittain, and William Shockley at
Bell Laboratories, it is fair to say that not even they would have imagined the proliferation
of and extent to which computing has reached. Yet, science and society continue to march
forward, looking for ever more computational power and faster processors. Before consid-
ering the future of computing however, we can obtain some perspective about the scope of
computers today through looking at the historical development of information processors.
Computers were not always silicon based nor made up of transistors. Rather, the earliest
processors were made up of vacuum tubes and electromechanical relays, physically taking up
large amounts of space. Arguably the first critical implementation of computers was during
World War II, with the British Colossus computers [1] used to break German wartime codes.
The war stimulated the scientific progression of digital computing and fortunately, scientists
responded to the challenge, helping decrypt intercepted Nazi transmissions.
Subsequently, new technological advances in transistors and integrated circuits changed
the classical computing landscape forever. Instead of a single bit of information taking an
individual vacuum tube, a solid-state chip only a tiny fraction of the volume of the vacuum
tube could hold millions of transistors, each representing a bit. Computers no longer needed

1
2 introduction

to take up entire floors of a building, but could even begin to become personalized for use in
the everyday home.
So how many bits can we fit into a microprocessor and how does information processing
scale? The well-known Moore’s law has predicted that the number of transistors which can
be placed onto an integrated circuit (IC) doubles approximately every two years [2]. The
trend has been traced for the past half century and demonstrates the ability of technology
to continue improving at exponential levels. Yet, there is a fundamental physical limit to
Moore’s law because as we continue to increase the density of bits, we eventually reach the
level of the individual atoms of silicon. At these scales, standard solid-state physics breaks
down, transitioning into the physics of the atomic scale. Specifically, quantum mechanics
begins to play a role: interactions between the atoms become no longer negligible, and
quantum tunneling between parts of the IC can occur. Already in our smallest present-
day processors, quantum mechanics is responsible for substantial gate leakage, resulting in
significant heating.
Therefore, in the terms of computing progress moving forward, there are two paths to
consider. The first is to understand what will be the fundamental limits to Moore’s law and
what techniques within classical computation and semiclassical solid-state engineering can
be done to continue improvement, even if not at Moore’s law levels. The second is to start
from quantum mechanics, perhaps even at the atomic level, and think about computing
and information processing by directly employing the quantum effects. The first path is
the task of electrical engineers, materials scientists, and computer engineers to figure out
different physical architectures for constructing ICs, improved materials to minimize loss
mechanisms while continuing to scale down, and shift towards more parallel processors
which will require more efficient and adapted computer programs. The second path has
resulted in the burgeoning field of quantum information processing, which we will motivate
in the next section. The experimental implementation in a solid-state system is the subject of
this thesis.

1.1 Computing with quantum mechanics

Devices which perform quantum information processing are called quantum computers. The
concept of quantum computing can be traced back to the early 1980s, first with the suggestion
by Richard Feynman for quantum mechanics simulations [3] and then for the solution of a
toy problem with a quantum algorithm developed by David Deutsch [4].
 1.1. computing with quantum mechanics 3

Feynman noted that classical computers would not be able to simulate quantum mechani-
cal systems efficiently. The general direction of quantum simulation using classical computers
is to describe the mean behavior of a system comprised of more than a million degrees of
freedom. However, in nuclear physics, atomic physics and chemistry, it is often important to
be able to simulate systems made up of tens to hundreds of quantum objects. In this case,
the mean field approach does not give a complete enough picture. Rather, it was suggested
that having control over quantum systems would permit the first principles construction of
many-body systems.
The first simple quantum algorithm was proposed by David Deutsch in 1985, using
quantum mechanics to solve essentially the problem of determining if a coin is fair or biased
more efficiently than any classical computing algorithm could [4]. But the proposed problem
was very limited in scope, and although Deutsch’s algorithm demonstrated a concrete way
in which quantum computers could beat a classical computer, it was not yet enough to
push forward with a major physical research effort to investigate and implement a quantum
computer.
The landscape of quantum information processing quickly changed, however, when Peter
Shor introduced an integer factoring algorithm which could exponentially outperform any
known classical computational algorithm [5]. The problem of factoring large numbers is in fact
very computationally difficult, with even the most complex classical computers requiring the
lifetime of the universe to complete the task. Interestingly enough, the factorizing problem
in reverse, integer multiplication, is very simply implemented with classical computers.
These two features, simplicity to multiply and the difficulty to factor, have led to the public-
key encryption scheme developed by Rivest, Shamir, and Adleman (RSA), widely used for
electronic business communication and transaction applications [6]. Furthermore, new
quantum information based encryption schemes were developed by Charles Bennett and
researchers at IBM [7]. Such quantum encrypted systems become unbreakable via classical
means, relying on the concepts of quantum entanglement and measurement. The possibility
that a quantum computer implementing Shor’s algorithm could be used for breaking one of
the most powerful classical encryption algorithms stimulated considerable interest in both
quantum computing theory and physical implementations to try to implement Shor’s or
develop new quantum encryption protocols. The combination of intellectual interest from
scientists in a variety of disciplines, and the realization that quantum computing might have
national security implications in the future, made it a topic of increasing importance.
Subsequently, in addition to a lot more quantum computing theory devoted towards
4 introduction

the development of new algorithms and novel applications of quantum information, there
was also a new theoretical emphasis on how to physically and experimentally implement a
quantum computer. The basic building block of such a quantum computer is the quantum bit
or qubit. It is similar to the classical bit in that it is a system comprised of two discrete states,
∣0⟩ and ∣1⟩. However, these states need to be any set of two quantum mechanical levels, such
as an electron spin or nuclear spin, or a pair of energy levels in an atom, ion or molecule. We
next briefly review some of the experimental realizations of quantum processors.

1.2 Experimental implementations of quantum processors

Building a quantum processor first requires a physical pair of quantum levels which are
addressable to form a qubit, the ability to couple multiple qubits, and a way to measure the state
of the qubits, all while maintaining quantum coherence, such that the quantum information is
not degraded and lost. Details about the various aspects of a quantum information processor
will be described later in this thesis in chapter 2.
Shortly following the discovery of Shor’s algorithm, the first successful experimental im-
plementation of quantum processors was realized using ensembles of nuclear spins in a single
molecule as the qubits [8]. The techniques of nuclear magnetic resonance (NMR), which
were already developed at a very high level for other applications such as magnetic resonance
imaging for medicine and chemistry, were easily transferred for performing operations on
the collection of spins. Another important property of NMR qubits was the ability to have
long coherence times (on the timescales of seconds) despite being composed of an ensemble
of spins. NMR quantum computers progressed very rapidly, moving from simple two-qubit
algorithms [8–10] up to ultimately a seven-qubit quantum computer capable of factoring
the number 15 and demonstrating the first experimental instance of Shor’s algorithm [11].
However, the scalability past seven qubits became very challenging as a result of increasing
complexity of experimental controls along with each qubit not being very ‘pure’ due to being
composed of a statistical distribution of molecular spins [12].
Another quantum computing experiment which matured very rapidly was trapped-ion
qubits, first proposed by Cirac and Zoller in 1995 [13]. The qubits are defined in the electron
or nuclear energy states of ions which are confined and trapped using electromagnetic fields.
Multiple qubits couple with one another through the collective motion of all the ions in the
trap, mediated via Coulomb interaction. The controls on each trapped-ion qubit and the
coupling of multiple qubits are performed via optical excitation using lasers. Here, again the
 1.2. experimental implementations of quantum processors 5

progress of trapped-ion quantum computing was very rapid owing to the strong experimental
foundations in atomic clocks and long coherence times [14] of ions. Currently, trapped-ion
quantum computers have demonstrated the ability to couple up to 8 calcium ions [15, 16].
There are also proposals involving the shuttling of ions between arrays of ion traps, and
chip-based trap schemes to scale the system further. Nonetheless, the increasing amount of
resources necessary to control a large-scale trapped-ion quantum computer is a daunting
challenge which will need to be addressed in its own right moving forward.
Although NMR and trapped ions have been relatively successful quantum processor
technologies, as we have alluded, the scalability and controls have still remained an out-
standing challenge. Another research approach has been solid-state quantum computing,
attempting to define and address the qubits on a chip, much like the transistors which are
now packed into an integrated circuit on a silicon microprocessor. In terms of qubits there
are solid-state approaches which aim to isolate single electron spins as in GaAs quantum
dots [17], nitrogen-vacancy centers in diamond [18, 19], and implanted phosphorous donors
in silicon [20] as well as approaches which use the collective quantum coherence of Cooper
pairs in superconducting tunnel junctions.
The benefits of solid-state approaches are the flexibility and volume of production which
current lithographic fabrication techniques provide. Technological development in electron
beam lithography has allowed for circuits to be defined with nanoscale precision. This type
of control over circuits allows for tailorable qubit energy levels as well as the possibility for
tunability in-situ. This is especially the case for the superconducting qubit architecture, which
uses macroscopic sized circuits to define the energy levels and coupling strengths of the qubits.
Here, the quantum mechanical states can be discrete Cooper-pair charge states on a type of a
superconducting tunnel junction known as a Josephson junction. The energy levels of the
superconducting qubit are tunable and tailorable via lithography of the Josephson junctions.
Another benefit of the superconducting qubit architecture is the all-electrical control using
standard microwave and radio-frequency (RF) engineering techniques. The well-developed
fabrication protocols and electrical controls could possibly allow for superconducting qubits
to be made in large numbers and have tailored and controllable properties.
Yet, in terms of real quantum processors, the superconducting qubit architecture has
lagged behind. The primary issue has been reduced coherence times. When the first super-
conducting qubits arrived on the scene around ten years ago [21], energy relaxation times
were on the order of nanoseconds. Recent progress has increased these times to the order
of micro-seconds. One standard goal in practice is for the probability of error when per-
6 introduction

forming a quantum operation to be very small, and below what is called the ‘fault-tolerant
threshold.’ Quantum computing theorists have placed this threshold at being able to perform
over ten-thousand operations before encountering a single error. When a qubit architecture
is capable of reaching this low error rate, there are a number of quantum error correcting
codes which can be enacted to make the quantum computer fault-tolerant. Whereas trapped-
ions and NMR systems have long coherence times making this threshold within reach, the
superconducting qubit architecture is still working to catch up.
Nonetheless, with the current state of the art, we will show, in this thesis, the ability to
perform simple quantum information processing on a quantum computer built with two
superconducting qubits. To some degree the results presented here help put the superconduct-
ing qubit architecture on the same map as other more developed quantum systems. Moving
forward, however, reaching the ultimate realization of a scaled-up quantum computer is still
a hefty challenge.

1.3 Overview of thesis

This thesis work demonstrates the first solid-state implementation of a quantum processor.
The qubits which we will work with are superconducting charge qubits, specifically the trans-
mon, which is a modified version of the Cooper-pair box. Coherence times of the transmon
qubit have now reached 1 − 2 µs setting up the possibility of the quantum information experi-
ments presented in this thesis. The architecture for the multi-qubit coupling will be circuit
quantum electrodynamics (QED), an on-chip version of cavity quantum electrodynamics
which is the fundamental interaction between a photon and an atom. We will see that this
architecture will allow us to use a separate quantum degree of freedom, namely the photons
in the cavity, to act as a quantum bus to mediate interactions between non-local qubits.
To be able to fundamentally understand the requirements of building a rudimentary
quantum processor, we will start this thesis with some of the basics of quantum information
processing (QIP) in chapter 2. This involves identifying a universal set of quantum gates,
including single-qubit and two-qubit gates, and how to concatenate them to construct simple
quantum algorithms to run on the processor. Chapter 2 will also describe the general quantum
state measurement process, including state tomography and entanglement quantification,
such that at the end of a set of quantum operations, we may identify the state of the system
and the degree of entanglement contained.
 1.3. overview of thesis 7

That will be followed by chapter 3, in which we will review superconducting qubits, and
especially describe the transmon qubit used in this work. There will also be discussion about
some of the basics of coupling to a microwave transmission line cavity in circuit QED. We
will be able to associate a number of key concepts from cavity QED, including the strong
and dispersive coupling regimes, which will be useful for quantum information processing.
Furthermore, there will be a discussion about the transmon qubit decoherence properties in
the circuit QED regime. Then, in chapter 4, we will describe how the language and concepts of
quantum information processing can be defined in our circuit QED system. We will provide
a description of how to build a quantum processor with transmon qubits in a microwave
cavity, understanding how to implement a universal set of gates. Details for how to generate
two-qubit entangling gates will be given, as well as a discussion which expands the idea of
the strong dispersive limit of cavity QED to a joint quantum state readout.
The experimental details about building up the quantum processor will be described
in chapter 5. We will review some of the sample fabrication details, including optical and
electron-beam lithography procedures, performed with the help of Luigi Frunzio, Blake
Johnson, and Joseph Schreier. We will also discuss considerations for designing the transmon
qubits and the microwave cavities. There will be a specific emphasis on the design of a qubit
with incorporated on-chip magnetic flux biasing (developed together with postdoc Johannes
Majer, and implemented with postdoc Leonardo DiCarlo). The whole experimental setup
from the chip-level up through the cryogenic circuitry and out to the room temperature
control electronics will also be described.
The next four chapters, chapter 6–chapter 9, will highlight experiments which progress
towards the implementation of quantum algorithms on our solid-state quantum processor.
First, in chapter 6 we describe experiments which point to a very good initialization of
the qubits to the ground state. Through a unique strongly-driven vacuum Rabi experiment,
we will characterize the average photon number of our microwave cavity, and translate that
to an equilibrium ground-state polarization of our qubit at the 99.99% level. Furthermore,
the chapter will also describe a number of metrics for characterizing single-qubit gates,
demonstrating gate fidelities of 99%, not yet reaching, but approaching the fault-tolerant
threshold. We will also highlight some preliminary work towards optimized pulse-shaping
to further reduce certain single-qubit gate errors.
Chapter 7 presents the first two-qubit quantum bus experiment, performed with Johannes
Majer, and shows the ability to reach both the strong and dispersive regimes of circuit
QED with two qubits. The coupling between two qubits via the cavity is demonstrated
8 introduction

spectroscopically via an avoided crossing and the presence of a ‘dark-state.’ We also describe
how this two-qubit coupling, which is a virtual-photon cavity-mediated two-qubit interaction,
can be used for coherent oscillations between states of the two qubits. These coherent swaps
represent a precursor for an entangling two-qubit gate.
Then, chapter 8 presents a new experiment performed together with Leonardo DiCarlo,
exploiting qubits with better coherence times and the ability to tune a novel two-qubit
coupling on and off with fast timescales. This new interaction is derived from the presence of
higher energy levels in the transmon charge-based qubits. Using on-chip magnetic flux bias
lines, the transition energies of the qubits are tunable, such that the two-qubit interaction can
be turned on and off at nanosecond timescales. This interaction is used to make an entangling
conditional-phase gate, permitting the generation of high fidelity two-qubit states, including
highly entangled two-qubit states. We further describe how the circuit QED architecture can
be used for determining these two-qubit states and characterizing the degree of entanglement
in our system.
Chapter 9 culminates with the implementation of two simple quantum algorithms on
our superconducting processor, again in work performed together with Leonardo DiCarlo.
Specifically, we describe how we program in the two-qubit Deutsch-Jozsa algorithm as well
as the four state Grover’s search algorithm, representing the first-ever solid-state quantum
processor.
Finally, chapter 10 will present some future directions for superconducting quantum
computing, specifically detailing anticipated experiments on three to four qubits.
 CHAPTER 2

Quantum Information Processing

Q
uantum computing, once merely a casual thought by a few notable scientists, including
Richard Feynman [3], in the 1980s, has blossomed into an interdisciplinary research
field encompassing wide areas of physics, computer science, and mathematics. Practical
aspects of realizing a physical quantum computing platform are now the subject of countless
research programs, with implementations spanning naturally occurring to man-made quan-
tum systems. As introduced in the previous chapter (chapter 1), this thesis will present in
detail the first solid-state demonstration of a simple quantum processor. However, before
delving into the physical system of circuit quantum electrodynamics (chapter 3 and chapter 4)
in which we realize such a processor, it is useful to review and understand the language of
quantum operations and algorithms for the sake of perspective and foundation.
Certainly, one could pick up a standard text on this subject, such as Nielsen and Chuang
[12], Mermin [22], or Kaye, Laflamme, and Mosca [23], to learn about all the nuances of
quantum information processing, from as simple as single-qubit operations to as complex as
Shor’s factoring algorithm and quantum error correcting codes. Such texts give a broad scope
of both the monumental prospects and challenges for making a quantum computer. Whereas
long range dreams of breaking RSA encryption and simulating real quantum systems are
worth keeping in the back of one’s mind for motivation, the practical quantum experimentalist

9
10 quantum information processing

has to start with building a quantum processor from the ground up and learn the basic
quantum algorithms and measurements for only a few qubits.
This chapter will describe quantum information processing on a more fundamental
level of quantum operations of a few qubits, picking relevant parts from the standard texts
mentioned previously. This will allow us to have a solid point of reference for the actual
experimental implementation to be described later in this thesis. We will start by describing
a set of single and two-qubit gates which form a universal set for computing (section 2.1,
section 2.2, section 2.3). Then we describe the general quantum computing process in terms
of building up simple two-qubit algorithms (section 2.4), including the Deutsch-Jozsa and
Grover’s search. Next, it is important to overview the quantum measurement problem and
how we can characterize a quantum state (section 2.5). Then, we demonstrate how to go from
simple state identification to the ability to measure the degree of entanglement in a system
(section 2.6). Finally, we end the chapter with a discussion about Bell inequalities and its role
in quantifying entanglement (section 2.7).

2.1 Universal quantum computing

In classical computing, the most basic unit of information is the bit, with two discrete states
0 and 1. Computational algorithms are comprised of binary logic operations, such as the
AND, OR, and NOT gates. The concept of universality refers to the ability to comprise any
computational algorithms with a closed set of simple gates [12]. For example, the NAND gate
and the NOR gate are each universal, such that using only combinations of each gate, one
can accomplish all basic binary logic operations which may be in an algorithm.
In quantum computing, instead of bits, we have qubits, which can be in not only the
discrete quantum states ∣0⟩ and ∣1⟩, but in fact arbitrary superposition states. Similar to uni-
versal logic operations, there also exists a set of quantum gates which are universal, such that
combinations of gates can realize complex quantum algorithms. However, unlike the classical
computational case where only a single gate is necessary, in the quantum case universality can
only be achieved with the combination of arbitrary single qubit gates and a two-qubit gate
such as the controlled-NOT (cNOT). The proof for this universality construction of quantum
computing is given in Ref. [24], showing that any unitary operation can be approximated to
arbitrary accuracy through a quantum circuit.
One of the key differences between the construction of a quantum computation and a
classical computation is reversibility. Classical gates such as the NAND, NOR, AND, and
 2.2. single-qubit gates 11

OR are destructive, or irreversible, in the sense that they take two inputs and return a single
output. However, reversible classical computing is certainly possible, and requires only a
function which takes an n-bit input to an n-bit output. Understanding reversible classical
computing is one way to step towards building a quantum computer, as quantum computing
is based upon the action of reversible unitary operations in quantum mechanics. For example,
the two-qubit gate cNOT is not only a unitary transformation within a two-qubit Hilbert
space, but also a two-bit reversible classical operation. Perhaps one of the most interesting
wrinkles is that although the cNOT is part of a universal set of gates for quantum computing,
it is not universal for classical reversible computation. Rather, it takes at the least a 3-bit
Toffoli gate or a ccNOT [25]. The reason that a quantum computer would require fewer
number of qubits per gate is the ability to generate entanglement and superposition between
qubits using certain gates, such as the Hadamard gate, or Hadamard combined with a cNOT.
These aspects will be explored in detail in the rest of this chapter.
The operations of a quantum computer can thus be summarized as the combination of
unitary operations on multiple qubits, and built up in a quantum circuit formalism [12]. The
operations on an n-qubit quantum circuit will be sequences of quantum gates, all of which
will be reversible transformations on the n-qubit register. Next, we build up this model of
quantum computing with the introduction of the most basic building blocks, the single-qubit
gates.

2.2 Single-qubit gates

Perhaps the simplest quantum operations to consider are those for just a single qubit. A single
qubit is comprised of only two quantum states, ∣0⟩ and ∣1⟩, and single-qubit gates traverse
through the Hilbert space spanned by these two states. We can represent a single qubit by
the state vector

∣ψ⟩ = a ∣0⟩ + b ∣1⟩ , (2.1)

with complex amplitudes a and b which are normalized ∣a∣2 + ∣b∣2 = 1. All single-qubit gates
can be represented as 2 × 2 unitary matrices. The space of such matrices are spanned by the
12 quantum information processing

z ∣⟩
∣ψ⟩
y

∣⟩−∣⟩ ∣⟩+∣⟩
√ √
 

∣⟩

Figure 2.1: The Bloch sphere. Geometrical representation of the state space of a single qubit
(two-level quantum system). The state of the qubit is represented by the Bloch vector, which is
a unit vector within the sphere, describe by two numbers, θ and ϕ.

identity (referred to as 1 or I in this thesis) along with the three Pauli matrices,

⎛0 1 ⎞
σx ≡ X ≡ (2.2a)
⎝ 1 0⎠
⎛0 −i ⎞
σy ≡ Y ≡ (2.2b)
⎝i 0 ⎠
⎛1 0 ⎞
σz ≡ Z ≡ . (2.2c)
⎝0 −1⎠

These Pauli matrices can be used to generate rotations about the x, y, and z axes to traverse the
entire two-qubit space, often pictorially represented by the Bloch sphere (shown in figure 2.1).
The rotation operations, which are also unitary gates, are given by

⎛ cos θ2 −i sin θ2 ⎞
R x (θ) ≡ X θ ≡ e −iθσx /2 = (2.3a)
⎝−i sin θ2 cos θ2 ⎠
⎛cos θ2 − sin θ2 ⎞
R y (θ) ≡ Yθ ≡ e −iθσ y /2 = (2.3b)
⎝ sin θ2 cos θ2 ⎠
⎛ e −iθ/2 0 ⎞
Rz (θ) ≡ Z θ ≡ e −iθσz /2 = , (2.3c)
⎝ 0 e iθ/2 ⎠

where θ is the angle of rotation.

We can combine the Pauli operators and the identity to form a generalized rotation about
 2.2. single-qubit gates 13

any axis n̂ given by

θ θ
R n̂ (θ) ≡ exp(−iθ n̂ ⋅ σ⃗/2) = cos 1 − i sin (n x σx + n y σ y + nz σz ) . (2.4)
2 2
Some important single-qubit gates are rotations of θ = ±π and θ = ±π/2, often referred to
as π-pulses and π/2-pulses, respectively. We can identify certain rotations with the Pauli
matrices, R y (π) = −iσ y , R x (π) = −iσx , Rz (π) = −iσz . Experimentally, it is often simpler to
access rotations about the three Cartesian axes and to use the set of rotation operators to build
up the more standard single-qubit gates which are used throughout the theoretical literature
and quantum computing texts. Specifically, quantum circuits often feature the single-qubit
gates such as the Hadamard, X gate, Z gate, phase gate, and π/8 gate, given by

1 ⎛1 1 ⎞
H=√ Hadamard gate (2.5a)
2 ⎝1 −1⎠
⎛0 1 ⎞
X= bit-flip gate (2.5b)
⎝ 1 0⎠
⎛1 0 ⎞
Z= phase-flip gate (2.5c)
⎝0 −1⎠
⎛ 1 0⎞
S= phase gate (2.5d)
⎝0 i ⎠
⎛1 0 ⎞
T= π/8 gate. (2.5e)
⎝0 exp(iπ/4)⎠

The Hadamard gate is very significant because it enables the qubit interference which is
necessary for many quantum algorithms. As we will show later in section 2.4, the Hadamard
gate allows one to access quantum parallelism, such that a single function may be evaluated for
a whole set of computational states at once. In terms of single-qubit rotations, the Hadamard
reflects a π/4 rotation around the y axis followed by a π rotation around the z axis.
The X gate, also commonly referred to as the NOT or bit-flip gate changes the compu-
tational basis value from one state to the other. It is equivalent to the σx Pauli operator, or
a rotation around the y axis by π. The Z gate, or the phase-flip gate, is simply the Pauli σz
operator and represents an azimuthal rotation of the Bloch vector by π. The phase gate can
be seen to be the square root of the Z gate, as S 2 = Z, and reflects an azimuthal rotation by
14 quantum information processing

π/2. The T or π/8 gate is the square root of the phase gate, T 2 = S. As noted in Ref. [12], this
is quite an unfortunate name given that it is a rotation of π/4 which enters.
Therefore, although a lot of the literature presents algorithms with a specific library of
single-qubit gates, in the end, they are all simply combinations of rotations about x, y and z,
which can be more easily accessible in particular experimental architectures. In Ref. [12] the
Hadamard, S, and T gates are part of the universal set for quantum computing. Here, we will
later show (chapter 4) that we can experimentally access the Cartesian rotation operators,
and we will use the appropriate combinations of such gates to eventually build up simple
algorithms (chapter 9). Furthermore, the specific set of rotations of π/2 about x, y, and z
generate the single-qubit Clifford group [26]. Later in this thesis we will discuss Clifford
group operations with regards to determining the average fidelity of single-qubit operations
(chapter 6).

2.3 Two-qubit entanglement gates

The previous section dealt with only single qubit logic. We can now expand to two qubits,
and investigate unique gates in this expanded Hilbert space which are not simply products
of single-qubit operations. One class of such gates are controlled operations. One qubit can
be labeled the control qubit and the other the target qubit. Controlled operations involve
an action on the target qubit which will change depending on the state of the control qubit.
Such two-qubit gates are the basis of generating entanglement and along with arbitrary
single-qubit rotations, complete the universal set (section 2.1) for approximating multi-qubit
unitary operations.

2.3.1 cNOT gate

The controlled-NOT or cNOT is the “canonical” two-qubit entanglement gate, used through-
out theoretical constructions of quantum algorithms as well as described in depth in standard
quantum computing texts [12, 22, 23]. The circuit representation for the cNOT gate is shown
in figure 2.2, with two inputs corresponding to the control and target qubits and then two
outputs. This two-in, two-out operation reflects a difference from a classical computing oper-
ation where two-bit gates end with a single output bit. For example, the classical analog of the
cNOT is the XOR gate, which takes two bits A and B and returns a single bit corresponding
 2.3. two-qubit entanglement gates 15

control qubit ∣A⟩ ∣A⟩

target qubit ∣B⟩ ∣A⟩ ⊕ ∣B⟩

Figure 2.2: Circuit representation for the controlled-N OT gate. In the quantum circuit
model, operations on different qubits are represented on different horizontal tracks. For
the cNOT gate, there are two qubits and two tracks, a control qubit along the upper track and
a target qubit along the lower track. In the cNOT gate, the control qubit is symbolized with a
solid black circle and the target qubit is symbolized with an open circle.

to the modulo 2 addition operator ⊕,

A⊕0 = A
(2.6)
A ⊕ 1 = 1 − A = Ā.

Classically, this is an irreversible process. However, the cNOT gate achieves a similar result,
but is reversible and describable by a unitary matrix. The action of the cNOT gate is to leave
the target qubit alone if the control qubit is in state ∣0⟩ and to flip the target qubit if the control
qubit is in state ∣1⟩. We can write this for two qubits as ∣A⟩ ∣B⟩ → ∣A⟩ ∣B ⊕ A⟩. Therefore,
cNOT can be written in a 4 × 4 unitary matrix representation with the columns and rows
being the computational basis states of two-qubits, ∣0, 0⟩, ∣0, 1⟩, ∣1, 0⟩, and ∣1, 1⟩ as

⎛1 0 0 0⎞
⎜ ⎟
⎜0 0⎟
=⎜ ⎟.
1 0
UCN OT ⎜ ⎟ (2.7)
⎜0 1⎟
⎜ 0 0 ⎟
⎝0 0 1 0⎠

We can recognize the difference between the cNOT and the XOR in that the cNOT is a
reversible operation, whereas the XOR actually has erased the information in the control bit,
leaving only a single bit of information in the target bit. However, it is also the association
of the cNOT with the classical XOR operation which makes it a ubiquitous reference in
quantum circuits, as it permits the possibility of transferring computation schemes written
for reversible classical computation over into the quantum language.
Therefore, given single-qubit gates and the cNOT two-qubit gate, we can start to ex-
plore more complex quantum algorithms through their concatenation in quantum circuits.
However, when we try to relate the quantum circuit formalism to a particular experimental
16 quantum information processing

implementation of qubits, the cNOT may or may not be the most natural selection for a
two-qubit primitive entangling gate. Rather, as practical quantum engineers, it is critical to
recognize the type of qubit interactions present, and then to employ the appropriate gate
which makes the most efficient use of resources. The formally solved quantum protocols and
algorithms which are simply broken down into single qubit unitaries and cNOT gates can
then be recompiled into the gates which are most easily accessible in a particular experimental
architecture.

2.3.2 c-Phase gate (cU i j )

A particular coupling that arises in many experimental architectures such as nuclear magnetic
resonance (NMR), as well as flux [27] and charge superconducting qubits [28], is the ZZ-
interaction, where the interaction Hamiltonian between qubit 1 and 2 is given by
ZZ
ZZ
E1,2
= σz ⊗ σz ,
(1) (2)
H1,2 (2.8)
4
where ⊗ is the outer product. This interaction thus corresponds to a unitary time-evolution
ZZ
given by U1,2 = exp[−iH1,2
ZZ
t]. Given the ability to turn on and off this interaction over a time
of t = ħπ/E1,2
ZZ
corresponds to

⎛ 1 0 0 0⎞
⎜ ⎟
⎜0 −i 0 0⎟
ZZ ⎜
= exp[iπ/4] ⎜ ⎟,
U1,2 ⎟ (2.9)
⎜0 0 −i 0⎟
⎜ ⎟
⎝0 0 0 1 ⎠

where we have used the computational basis states ∣0, 0⟩ [with corresponding vector (1, 0, 0, 0)],
∣0, 1⟩ [with corresponding vector (0, 1, 0, 0)], ∣1, 0⟩ [with corresponding vector (0, 0, 1, 0)],
and ∣1, 1⟩ [with corresponding vector (0, 0, 0, 1)] where ∣n1 , n2 ⟩ denotes excitation level n1 on
qubit 1 and n2 on qubit 2. This unitary operation can be combined with rotations around z of
each qubit, Rz (−π/2) and Rz (π/2), so that we arrive (up to a global phase factor) at the
(1) (2)
 2.3. two-qubit entanglement gates 17

control qubit
⎛   ⎞
⎜   ⎟
 ⎜ ⎟
⎜   ⎟
⎝   −⎠
target qubit
Figure 2.3: Circuit representation for the controlled-Phase gate. In the c-Phase gate, also
commonly labeled as cU i j , both the control and target qubit are symbolized with a solid black
circle, and the specific computational basis state which picks up the −1 phase shift is written to
the side as i j. In the case shown here i j = 11.

conditional-phase or c-Phase gate,

⎛1 0 0 0⎞
⎜ ⎟
⎜0 0 0⎟
= exp[iπ/4] ⎜ ⎟.
( j) 1
cU11 = [Rz (−π/2) ⊗ Rz (−π/2)] U i,ZZj
(i)
⎜ ⎟ (2.10)
⎜0 1 0⎟
⎜ 0 ⎟
⎝0 0 0 −1⎠

This particular c-Phase gate corresponds to a phase shift of π on the target qubit excited
state when the control qubit is in the excited state ∣1⟩. The circuit representation is shown in
figure 2.3. Through manipulating the rotation around z of either qubit, we can form any of
the three other c-Phase gates as well,

⎛1 0 0 0⎞ ⎛ 1 0 0 0⎞ ⎛−1 0 0 0⎞
⎜ ⎟ ⎜ ⎟ ⎜ ⎟
⎜0 1 0 0⎟ ⎜0 −1 0 0⎟ ⎜0 0⎟
cU10 = ⎜ ⎟, ⎜ ⎟, cU00 = ⎜ ⎟,
1 0
⎜ ⎟ cU01 = ⎜ ⎟ ⎜ ⎟ (2.11)
⎜0 0 −1 0⎟ ⎜0 0 1 0⎟ ⎜0 0⎟
⎜ ⎟ ⎜ ⎟ ⎜ 0 1 ⎟
⎝0 0 0 1⎠ ⎝0 0 0 1 ⎠ ⎝0 0 0 1⎠

reflecting the control qubit state being ∣0⟩ and then also swapping the roles of the control
and target qubits.
The cNOT gate and c-Phase gate are intimately related, differing by only single-qubit
rotations. The cNOT can be built (see figure 2.4) from the c-Phase with Hadamard gates on
the target qubit,

U cNOT = H (2) cU11 H (2) . (2.12)

Therefore, although many quantum algorithms are written in terms of cNOT operations
as the two-qubit operation, it is not too difficult to translate these sequences in terms of c-
18 quantum information processing

control qubit

= 11
target qubit H H

Figure 2.4: Circuit form for constructing a cNOT from a c-Phase gate. A cNOT gate can
easily be constructed from the c-Phase gate cU11 by performing single-qubit Hadamard gates
on the target qubit before and after.

Phase along with single-qubit rotations. Examples are the Grover’s search and Deutsch-Jozsa
algorithms (section 2.4) discussed later in this chapter. Ref. [22] refers to the c-Phase gate in
fact as a more natural and efficient gate compared to cNOT and we will find that in the circuit
QED charge qubit architecture which is the experimental focus of this thesis, the c-Phase
will be the two-qubit gate of choice (section 4.3.3).

√
2.3.3 iSWAP and iSWAP gates
Another interaction scheme which arises quite frequently in experimental quantum comput-
ing implementations is the XY or transverse qubit-qubit coupling. The relevant interaction
Hamiltonian is given by
XY
XY
E1,2
= (σx σx + σ y σ y ) ,
(1) (2) (1) (2)
H1,2 (2.13)
4
and often written in terms of Pauli raising and lower operators,
XY
XY
E1,2
= (σ+ σ− + σ− σ+ ) .
(1) (2) (1) (2)
H1,2 (2.14)
2
This type of coupling can be realized in quantum dot spins [17], nuclear spins interacting via a
two-dimensional gas [29], as well as Josephson charge qubits coupled either by a transmission
line resonator [30] or other Josephson junctions.
The time-evolution of the two-qubit system due to this type of coupling does not simply
result in a controlled operation, such as cU i j or cNOT. Instead, it is most suited for generating
√
the iSWAP and iSWAP two-qubit gates, which can also form part of a universal set when
combined with appropriate single-qubit rotations.
 2.3. two-qubit entanglement gates 19

control qubit
= iSWAP iSWAP
target qubit

control qubit
= iSWAP iSWAP
target qubit

√
Figure 2.5: Circuit form for constructing a cNOT from an iSWAP or iSWAP gate. When
the accessible two-qubit √interaction is XY and not ZZ, the natural two-qubit entangling gates
are either the iSWAP or iSWAP. Creating a cNOT gate then requires the concatenation of at
least two of each of the gates, combined with multiple single-qubit rotations around various
directions.

By turning on the XY interaction for a time t = πħ/E1,2

XY
, we arrive at the iSWAP operation,

⎛ 1 0 0 0 ⎞
⎜ ⎟
⎜ ⎟
U iSWAP = exp [−iH1,2 XY ] = ⎜ ⎟.
XY π 0 0 i 0
⎜ ⎟ (2.15)
E1,2 ⎜ ⎟
⎜ 0 i 0 0 ⎟
⎝ 0 0 0 1 ⎠

One cannot construct a cNOT from just simple single-qubit rotations along with a single
iSWAP. However, if we are allowed to use two iSWAP gates, then we do have this possibility,

U cNOT = [1(1) ⊗ R x (π/2)] [Rz (−π/2) ⊗ Rz (π/2)] U iSWAP

(2) (1) (2)

(2.16)
[R x (π/2) ⊗ 1(2) ] U iSWAP [1(1) ⊗ Rz (π/2)] ,
(1) (2)

and shown in quantum circuit form in figure 2.5.

Similarly, the same XY interaction can be turned on for half of the time t = πħ/2E1,2
XY
,
20 quantum information processing

√
which then gives rise to the iSWAP gate

⎛ 1 0 0 0 ⎞
⎜ √ √ ⎟
⎜ ⎟
U iSWAP = ⎜ ⎟.
0 1/ 2 i/ 2 0
√
⎜ √ √ ⎟ (2.17)
⎜ ⎟
⎜ 0 i/ 2 1/ 2 0 ⎟
⎝ 0 0 0 1 ⎠
√
Again, just as for the iSWAP, it takes two iSWAP gates to construct a cNOT gate along
with single-qubit rotations,
√
U cNOT =e iπ/4 Rz (π/2)R n⃗1 (π/3)R n⃗2 (π/3) iSWAP
(2) (1) (2)

√
Rz (π) iSWAP[R y (π/2) ⊗ R y (π/2)]Rz (−π/2)
(1) (1) (2) (2)

√ √
where n⃗1 = (1, 1, −1)/ 3 and n⃗2 = (−1, 1, 1)/ 3.
√
The constructions of cNOT in terms of iSWAP and iSWAP can get quite expensive in
terms of the time it takes to perform all the operations. Although single-qubit rotations are
relatively simple to implement, at present they certainly take up a non-trivial fraction of the
relaxation lifetime of the qubit. In addition, the recipes above require two copies of either
√
iSWAP or iSWAP. The time it takes to perform either gate is dependent on the interaction
strength and the ability to turn the interaction on and off very rapidly. Depending on the
qubit architecture, implementing longer gate sequences, which is necessary for performing
quantum algorithms, will require a careful economy of the total gates used, both single-qubit
rotations and entangling gates.

2.4 Quantum algorithms

With access to a universal set of quantum gates, we can now construct algorithms which
exploit superposition and entanglement to perform specific computations. Here, we will be
able to see how quantum computers can theoretically solve certain problems more efficiently
than classical computers. To get a feel for how such quantum algorithms can be built, we can
first investigate how a quantum computer can be programmed to evaluate some function
f (x) for multiple values of x simultaneously, or what is known as quantum parallelism. Then,
we move on to discuss a few quantum algorithms which are implementable in the most basic
quantum processors made up of only two qubits.
 2.4. quantum algorithms 21

2.4.1 Quantum parallelism in an algorithm

Suppose we start with a register of n + m qubits, where n qubits can be thought of as control
qubits on which we can give a specific set of inputs, to then perform a computation on m
target qubits which will be the output. The separate registers allow for reversibility of the
computation. Now suppose we have an (n + m) × (n + m) unitary transformation U f in
which we encode a function f (x) such that its action on any computational basis state gives

U f (∣x⟩n ∣y⟩m ) = ∣x⟩n ∣y ⊕ f (x)⟩m , (2.18)

where x and y are n and m bit integers and ⊕ represents bitwise addition mod 2. By starting
with the output register of qubits in ∣0⟩ it is possible to evaluate f (x) and have the result in
the output register,

U f (∣x⟩n ∣0⟩m ) = ∣x⟩n ∣ f (x)⟩m . (2.19)

However, now we can employ the ability to produce superpositions of each qubit to operate
U f only after applying a Hadamard transformation on all the input qubits. The n-qubit
Hadamard gives the maximal superposition state of the full register. For example, with two
qubits,
1 1
(H (1) ⊗ H (2) )(∣0⟩ ∣0⟩) = √ (∣0⟩ + ∣1⟩) √ (∣0⟩ + ∣1⟩)
2 2
1
= (∣00⟩ + ∣01⟩ + ∣10⟩ + ∣11⟩)
2
1
= (∣0⟩2 + ∣1⟩2 + ∣2⟩2 + ∣3⟩2 ) (2.20a)
2
giving a maximal superposition state involving all the computational states in the 2-qubit
input register. Therefore, now by applying U f after the n-qubit Hadamard, H ⊗n = H (1) ⊗
H (2) ⊗ ... ⊗ H (n) , on the ground state of the n-qubit register, we find
1
U f (H ⊗n ⊗ 1m ) ∣0⟩n ∣0⟩m = √ ∑ ∣x⟩n ∣ f (x)⟩m , (2.21)
2n x
which now contains all evaluations of the function f , despite having only operated f once by
applying U f .
Note that the problem remains in how to access all this information about f . It is possible
to measure in the computational basis states of each of the individual qubits. Yet, to simply
22 quantum information processing

H or

H Uf Up M or
...
...

...
H or

encode function measure

create
process
superposition
Figure 2.6: Anatomy of a quantum algorithm. The general quantum algorithm can be broken
down into four main parts. The first stage takes an initialized register of qubits into a superpo-
sition state via single-qubit operations, such as Hadamard gates. Then, a multi-qubit function
f is applied through a multi-qubit unitary operation which can be comprised of single-qubit
and two-qubit gates. Next, a processing step also made up of single-qubit and two-qubit gates
is performed to allow interpretation of the qubit register in the computational basis. Finally,
measurements are performed on all or some of the qubits.

measure all of the m qubits in the output register would only randomly reveal with equal
probability some choice of x0 < 2n . Therefore, we would only find out about the function
f (x0 ) at a particular random x0 . In this case, we have only performed what a classical
computer could easily have done.
However, quantum algorithms often employ a further stage applying additional unitary
gates which serve to form relationships between multiple evaluations of f for different values
of x. Here, to be able to know the values of certain combinations of f also means losing
the ability to know about individual values of f (x). A classical computer could only give
the values of relationships by making all of the individual independent evaluations. The
advantage gained through quantum algorithms is through the quantum mechanical concept
of interference, being able to tradeoff one kind of information for another.
Therefore, we can summarize the general computation structure with primarily four
stages of a quantum algorithm: the register of qubits must be placed into a superposition;
then a unitary function which encodes a function is applied; next a processing step to transfer
relational information into a form which can be readout; and finally a multi-qubit state
readout. These steps are summarized in figure 2.6 in a quantum circuit model.
 2.4. quantum algorithms 23

2.4.2 Deutsch-Jozsa algorithm

One of the simplest demonstrations of how a quantum algorithm can outperform a classical
computation is with the Deutsch-Jozsa (DJ) algorithm [31]. It represents a situation with a
quantum tradeoff of one particular information for a different global information. It is also
an example of quantum phase kick-back, where the target qubit can be placed in an eigenstate
of a controlled unitary such that an eigenvalue can be associated with the state of the control
register.
The DJ algorithm solves Deutsch’s problem [4] which can be succinctly described as
finding out whether a function f which takes a single bit x to a single bit f (x), is constant or
balanced. There are only four possible functions that take one bit to one bit, two of which
are constant, f (x) = 0, f (x) = 1, and two of which are balanced, f (x) = x, f (x) = 1 − x.
Another way of motivating the problem is to consider the case of finding out whether a coin
is biased (constant) or fair (balanced). There are two possible fair coins, with heads and tails
on each side, and two possible biased coins, with two heads or two tails.
Classically Deutsch’s problem is solved by querying f (x) for all values of the input bit
x = 0, 1, or looking at both sides of the coin. However, the DJ algorithm will allow us to
avoid finding out the information of specifically f (x = 0) or f (x = 1), in favor of knowledge
about the nature of the function, as balanced or constant. This is especially relevant when
evaluating f might be very computationally expensive.
Consider a unitary transformation which applies one of the four possible functions,
operating on two qubits, one of which serves as a control qubit, and the other as the target
qubit,

U f (∣x⟩ ∣y⟩) = ∣x⟩ ∣y ⊕ f (x)⟩ . (2.22)

The two constant functions are reflected by always giving 0 or always giving 1 regardless
of the input bit. These functions are implemented using U f 0 = 1 ⊗ 1 and U f 1 = 1 ⊗ X, for
results 0 and 1, respectively. The two balanced functions are reflected by always returning the
same value, such that f (x) = x, or always returning the opposite value, where f (x) = 1 − x.
These functions are applied using U f 2 = cNOT, and U f 3 = z-cNOT, respectively. The unitary
24 quantum information processing

matrix corresponding to z-cNOT is

⎛0 1 0 0⎞
⎜ ⎟
⎜1 0⎟
Uz-cNOT = ⎜ ⎟,
0 0
⎜ ⎟ (2.23)
⎜0 0⎟
⎜ 0 1 ⎟
⎝0 0 0 1⎠

where the state of the target qubit is flipped when the control qubit is in the state ∣0⟩.
The DJ algorithm is given in quantum circuit form as in figure 2.7. The algorithm is very
simple. Let qubit 1 be the control and qubit 2 be the target. We first apply rotations around
the y axis of π/2 on the target qubit and −π/2 on the control qubit to start, placing them into
√ √
the superpositions ∣ψ⟩1 = (∣0⟩ + ∣1⟩)/ 2 and ∣ψ⟩2 = (∣0⟩ − ∣1⟩)/ 2. That is then followed by
the application of any of the four unitary transformations that implement either a balanced
or constant function.
For illustrative purposes, suppose we apply the identity unitary U f 0 = 1 ⊗ 1 for one of the
constant functions. In this case, the two superpositions remain the same. The processing step
consists of applying a R y (−π/2) rotation on the target qubit and a R y (π/2) rotation on the
control qubit, which in the case nothing is done in the unitary transformation stage, simply
undoes the original superpositions created and return the register to the initial states.
However, suppose we instead have a balanced function, such that the unitary is U f 2 =
cNOT. An interesting thing occurs in this case, as the superposition state of qubit 2 is actually
an eigenstate of cNOT, with an eigenvalue of −1, no matter the state of qubit 1:

∣0⟩ − ∣1⟩ ∣0⟩ − ∣1⟩

cNOT ∣x⟩ ( √ ) = (−1)x ∣x⟩ ( √ ) . (2.24)
2 2
This can be interpreted as the eigenvalue giving a phase ‘kick back’ to the control qubit. Here,
an entangled state of the two qubits is not an intermediary step, but rather the entangling gate
serves to return a phase onto the control register. Therefore, in the case of the DJ algorithm,
√
the original positive superposition on qubit 1, ∣ψ⟩1 = (∣0⟩+∣1⟩)/ 2, gets turned into a negative
√
superposition, ∣ψ⟩1 = (∣0⟩ − ∣1⟩)/ 2. Then, the final set of rotations in the processing step will
place the control qubit into the state ∣1⟩ and allow the target qubit to return to the state ∣0⟩.
A final measurement of the state of the control qubit is in fact all which is needed at the
end of the algorithm. From the state being either ∣0⟩ or ∣1⟩, we are able to discern whether the
applied unitary corresponds to a balanced or constant function. Throughout the algorithm,
 2.4. quantum algorithms 25

control or

Uf
target
M or

Figure 2.7: Deutsch-Jozsa algorithm. The two qubits are initialized in their ground states. The
first superposition step involves only single-qubit rotations. The functions, either constant or
balanced, are encoded through applying the appropriate two-qubit unitary. The processing stage
involves only single-qubit rotations, serving to rotate the qubits to either ∣0⟩ or ∣1⟩ depending
on the form of the function. The final measurement need only be performed on the control
qubit to determine the nature of the function.

the target qubit simple goes along for the ride, providing the quantum phase kick-back
necessary in the case of the balanced functions.
With the DJ algorithm, we thus extract information about f (0) + f (1), instead of finding
out specifically what f (0) or f (1) are, which would be necessary for any classical computation
process. A classical algorithm would have required two calls to the function to determine the
flavor of the function, whereas here we can succeed deterministically with a single call. Note
that the DJ algorithm can be extended to more qubits for functions which deal with inputs
greater than a single bit. In this case, it also remains a deterministic algorithm requiring a
single call to the multi-qubit unitary, so long as the entire space of functions can be split into
either balanced or constant functions. Chapter 9 will demonstrate the implementation of the
DJ algorithm in our superconducting circuit QED processor.

2.4.3 Grover’s search algorithm

Another class of algorithms are for search with a quantum computer. Consider the problem
of searching for a specific name in an unordered database of N names. Classically, the best
that one can do is to choose names sequentially or at random, a process which takes on
average N/2 queries. However, Grover’s quantum search algorithm [32] can give a polynomial
speed-up on the classical case. Specifically, for searching an n-bit integer out of a list of N = 2n
different integers, Grover’s algorithm does better than random classical queries by a factor of
√
1/ N.
Suppose we start with log2 N qubits, and represent each of the database entries with a
ket, {∣1⟩ , ∣2⟩ , ..., ∣N⟩}. Next, we can imagine a quantum oracle which will be a black box for
26 quantum information processing

performing the search over the N entries. We can think of the oracle as applying a unitary
transformation U O to the entire qubit register, with the properties that

U O ∣α⟩ = − ∣α⟩ (2.25a)

U O ∣x⟩ = ∣x⟩ , for all x ≠ α. (2.25b)

The transformation thus has the effect of marking with a phase the specific entry ∣α⟩, which
is the entry we are searching for from the entire database. The processing of the search
is performed with the application of another unitary transformation U ϕ which induces a
conditional phase shift on every single state except for the first state, ∣1⟩,

U ϕ ∣1⟩ = ∣1⟩ (2.26a)

U ϕ ∣x⟩ = − ∣x⟩ , for all x ≠ 1. (2.26b)

Grover’s algorithm is then implemented as follows:

• Create an equal superposition state, by applying the n-qubit Hadamard, H ⊗n :

1 N
∣s⟩ = √ ∑ ∣x⟩ . (2.27)
N x=1
√
• Perform a Grover iteration R ≈ π 2n /4 times, where a Grover iteration involves

1. Apply the oracle transformation U O .

2. Apply the n-qubit Hadamard, H ⊗n
3. Apply the conditional phase transformation U ϕ
4. Apply the n-qubit Hadamard, H ⊗n

• Measure the final qubit register, which should give the computational basis state we
are searching for ∣α⟩.

In terms of the generalized quantum algorithm, the Grover iteration (figure 2.8) contains
both the function encoding part as well as the processing. However, it must be repeated
multiple times due to the nature of the algorithm, which can be understood as a routine which
turns a phase into a detectable amplitude. With only conditional phase transformations and
n-qubit Hadamard gates, multiple Grover iterations serve to amplify the amplitude of the
target state each time.
 2.4. quantum algorithms 27

n
qubits G G G M

G =

Figure 2.8: Grover’s search algorithm. The algorithm consists of an initial n-qubit Hadamard
H ⊗n for generating a full superposition of all the qubit states. The rest of the algorithm requires
repeating the Grover iteration G until the amplitude of the searched state is ∼ 1. The Grover
iteration consists of the application of an oracle unitary function U O which performs the search,
followed by processing steps involving H ⊗n and a conditional phase transformation U ϕ , whose
properties are discussed in the main text.

The first part of the Grover iterate, the multi-qubit oracle U O does all of the encoding
work, by marking the phase of the searched for target state. If such a phase could easily be
detected then the entire search problem would be finished. Unfortunately, to distinguish that
phase requires the processing stages of the other three parts of the Grover iterate, which we
will call U⊥ = H ⊗n U ϕ H ⊗n . This operator can be written as (2 ∣s⟩ ⟨s∣ − 1). We can understand
what the operator does by applying it to an arbitrary superposition state ∣ϕ⟩ = ∑x a x ∣x⟩ where
the mean of the amplitudes is given by m = ∑x a x /N. In this case,

U⊥ ∣ϕ⟩ = ∑(m − α x ) ∣x⟩ , (2.28)

x

which represents returning a superposition state that has mean (N − 1)m, by flipping the
amplitude of all the states around the mean m.
It is precisely this mean inversion which allows an amplitude amplification of the target
state ∣a⟩ in the Grover iteration. We can use a picture to represent the action of the Grover
iteration. In figure 2.9a, we start off with having applied the first H ⊗n , taking us to the
√
equal superposition state with all basis states having the same amplitude of 1/ N. The
28 quantum information processing

mean value (a)

∣a⟩ target state

UO
(b)

U⊥ ∣a⟩ (c)

Figure 2.9: Cartoon state illustration of Grover iteration. The search space starts off in an
equal superposition of all states in the qubit register Hilbert space as in (a). The appropriate
oracle U O for finding target state ∣a⟩ is applied, inverting the phase of only ∣a⟩.

√
mean amplitude value of all the states here is N and indicated by the dashed line. Next in
figure 2.9b, the oracle U O has flipped the sign of the target state ∣a⟩, which slightly lowers
the overall mean. Then, applying U⊥ will invert all the states about the mean, increase the
size of the amplitude on ∣a⟩ while diminishing the amplitudes on all the rest of the states
(figure 2.9c). Now, repeating the application of U O followed by U⊥ will continue to push the
overall mean down and increasing the size of the amplitude of ∣a⟩. It can be shown that it
√
then takes ≈ π N/4 repetitions to obtain an amplitude for ∣a⟩ of ≈ 1.
For the simplest search of only 4 entries using a two-qubit register, it is possible to perform
the Grover’s search algorithm to find the target state through only 1 iteration. This has been
implemented in NMR, linear optics, and trapped-ion quantum computer implementations.
In chapter 9 of this thesis, we will present the first implementation of such an algorithm with
superconducting qubits and go into a step-by-step breakdown of its operation.
 2.4. quantum algorithms 29

2.4.4 Shor’s and other quantum algorithms

Besides the DJ and Grover’s algorithms, there are a number of other quantum algorithms that
similarly exploit properties such as phase kick-back and amplitude amplification. We group
the rest of these algorithms in this subsection, without going into detail into any of them, as
they represent a class of quantum computing beyond the scope of this experimental thesis.
Nonetheless, it is still important to recognize for completeness and future motivation the exis-
tence of more complex algorithms which might even lie on the horizon for superconducting
quantum processors.
Many of the more complex quantum algorithms are built up from the quantum Fourier
transform (QFT), which is defined as the map
n
1 2 −1 x
∣x⟩ → √ ∑ e 2πi 2 n y ∣y⟩ . (2.29)
2n y=0

The QFT can be performed on an n-qubit register through a decomposition into only
Hadamard gates and conditional-phase gates. By operating the QFT on a superposition
of quantum states, we effectively apply the classical discrete Fourier transform to all 2n input
states in parallel. A full treatment of the QFT can be found in Ref. [23]. The QFT is applied in
various algorithms for estimating mathematical quantities, providing exponential speed-ups
over classical algorithms. For example, it is used for estimating eigenvalues of a unitary oper-
ator using the quantum phase estimation algorithm, as well as for finding discrete logarithms
[23].
Perhaps the most well-known quantum algorithm which employs the QFT is Shor’s
algorithm for factorization of a number N into prime numbers. Shor’s algorithm consists
of two primary phases, the first phase being a translation of the factoring problem into a
problem of finding the period of a function, and second phase using the QFT for finding
the period. The exponential speed-up occurs during the second quantum phase. Again,
details about both of these stages can be found in any of the listed quantum information texts
[12, 22, 23].
The discovery of Shor’s algorithm in 1994 actually represented a serious historical para-
digm shift in regards to experimental efforts for quantum computing. The primary use of
factoring large numbers is in fact for breaking the very widely used public-key encryption
scheme of RSA. RSA is a very ubiquitous protocol for cryptography which relies on the
difficulty for classical computers for factoring large numbers. Shor’s algorithm showed that it
30 quantum information processing

could be broken efficiently using a quantum computer. Subsequently, the quantum computers
gained a lot of visibility, pushing forward numerous experimental efforts.
Shor’s algorithm is in fact the most complex algorithm to have been implemented in an
experimental quantum processor. Using an NMR system of seven qubits, researchers at IBM
Almaden Laboratory managed to factor 15 into 5×3 [12]. With the superconducting two-qubit
processor described in this thesis, we cannot yet implement Shor’s algorithm. However, the
further development of the superconducting qubit architecture will hopefully lead to this
possibility.

2.5 Quantum measurement

At the end of performing the operations which comprise a quantum algorithm on a qubit
register, the final step is a quantum measurement of the register, by which we gain access to
information about the underlying quantum state, and hence the result of the computation.
Measurements can be considered to have an associated observable, which is Hermitian and
has real eigenvalues with corresponding eigenkets to span the state space. Historically, the
action of measurement has been a sensitive issue, with regards to how a classical macroscopic
channel can be used to infer microscopic quantum states.
In the earlier days of quantum mechanics, the Copenhagen interpretation presented
quantum measurement as wavefunction collapse. For a single qubit in a superposition state,
∣ψ⟩ = α0 ∣0⟩ + α1 ∣1⟩, a measurement of the qubit projection onto state ∣0⟩, P = ∣0⟩ ⟨0∣ will
return the qubit in the state ∣ψ⟩ = ∣0⟩ with probability ∣α0 ∣2 and in the state ∣ψ⟩ = ∣1⟩ with
probability ∣α1 ∣2 = 1 − ∣α0 ∣2 . Here, it is the act of measurement which forces the state into one
of the two eigenbasis states of P.
Although numerous thought experiments [33] have challenged the ideas of this awkward
measurement formalism, actual experimental progress in quantum information has led to
real measurements of quantum mechanics. Such experiments do not have perfect projective
measurements; rather, there can be statistical noise which results in the incorrect identification
of a measurement result. Therefore, it becomes crucial to obtain measurement statistics
on starting quantum mechanical states and a completely general framework for describing
quantum measurements has been developed, known as the Positive Operator-Valued Measure
(POVM) formalism [12]. This takes into account the possibility of weak (non-projective)
measurements, as well as statistical noise on the measurement process, allowing for the
probability of misidentifying one basis state as another. POVMs are related to the statistical
 2.5. quantum measurement 31

treatment of the state vector describing a quantum system, which is the density matrix
formalism.
For the results presented in this thesis, we only deal with ensemble-averaged measure-
ments, giving a simpler version of measurement theory. Specifically, by repeating the two-step
process of preparation of a quantum state and then performing a subsequent measurement,
we obtain expectation values of the form tr(ρM) where ρ is the density matrix of the quan-
tum state and M is a Hermitian measurement operator. The problem thus becomes one of
identifying what is the the measurement operator corresponding to the system, and then to
use ensemble measurements of the state to identify components of the state in a technique
known as quantum state tomography. We first start with a description of representing the
state using the density matrix formalism.

2.5.1 Density matrix

The density matrix representation can be a more powerful tool than the standard wavefunc-
tion formalism for real experimentally generated quantum states. It is especially useful for
describing ensembles of quantum states, for example, a qubit coupled to an environmental
bath, or a system of multiple qubits. Furthermore, processes such as relaxation and decoher-
ence can lead to statistical mixtures of pure quantum states, and can be accounted for with
non-equilibrium time-evolution of the density matrix through master equations [34].
Suppose a quantum system is comprised of many states ∣n⟩, with probability p n of being
in each. Then, the most general density matrix of the system is given by

ρ ≡ ∑ p n ∣n⟩ ⟨n∣ , (2.30)

n

where the sum is performed over all states n and ∑n p n = 1. Similar to the state vector, the
density matrix evolves under unitary transformations,

ρ → ∑ p n (U ∣n⟩)(⟨n∣ U † ) = U ρU † . (2.31)
n

Furthermore, this formalism provides a classification of the types of quantum states produced.
If the state of the system is known exactly to be describable by a state vector ∣ψ⟩, then it is a pure
state, with a density matrix given by ρ = ∣ψ⟩ ⟨ψ∣. If only partial information about the state is
known, then the system can be described by a mixed state, or a statistical ensemble of the
pure states. As the wavefunction formalism is limited to only describing a state which is pure,
32 quantum information processing

in the case of a mixed state we use the density matrix as defined in (2.30). The probabilistic
mixing of pure states can arise due to noise processes such as relaxation, decoherence, or
heating.
Some properties of the density matrix which are useful to keep in mind are

1. tr(ρ) = 1.

2. ρ is Hermitian.

3. ρ is always a positive operator, such that for any state ∣ϕ⟩, we have ⟨ ϕ ∣ ρ ∣ ϕ ⟩ ≥ 0.

4. The full joint density matrix of separable individual systems is the tensor product of
the individual density matrices, ρ1 ⊗ ρ2 ⊗ ... ⊗ ρ n .

5. A pure state has Tr(ρ2 ) = 1. A mixed state has Tr(ρ2 ) < 1.

Specifically with regards to mixed states of single qubits, the density matrix representation
allows one to see that there is also a Bloch sphere, described by a Bloch vector ⃗r , where

1 + ⃗r ⋅ σ⃗
ρ= , (2.32)
2
and ⃗r is now a real three-dimensional vector with ∣⃗r ∣ ≤ 1. A pure state will have ∣⃗r ∣ = 1 whereas
a mixed state will be a vector within the interior of the Bloch sphere.
The density matrix formalism is also a good way to represent ensemble measurements in
a quantum system. For example, the expectation value of the operator A, can be written as

⟨A⟩ = ∑ p n ⟨ ψ n ∣ A ∣ ψ n ⟩ = tr(ρA). (2.33)

n

One other feature of the density matrix of composite systems is the ability to describe
a subsystem through a partial trace. Namely, if we have a quantum state comprised of two
systems A and B, described by ρ AB , then the average properties of subsystem A can be
represented by a density matrix,

ρ A = trB (ρ AB ) (2.34)

where trB reflects a partial trace over the elements of subsystem B. This partial density matrix
formalism can be especially useful in the case of entangled quantum systems, where there is
no way to associate a pure wavefunction state to the subsystem A.
 2.5. quantum measurement 33

Section 2.5.2 will demonstrate how combinations of ensemble measurements can actually
be used to retrieve the density matrix of the quantum system with a technique known as
quantum state tomography.

2.5.2 State tomography

Quantum state tomography involves finding the elements of the density matrix which repre-
sents an unknown quantum state. Given a single copy of ρ it is impossible to fully determine
the state. But we can estimate ρ by preparing ensembles of the same state and then measuring
a whole host of observables.
For a single qubit state, the density matrix ρ can be written in terms of the orthonormal
set of Pauli operators, {σx , σ y , σz } along with the identity, 1 as

tr(ρ)1 + tr(σx ρ)σx + tr(σ y ρ)σ y + tr(σz ρ)σz

ρ= , (2.35)
2
which is simply an expansion of (2.32) using the components of the Bloch vector [12].
The trace expressions, i.e. tr(σx ρ), are simply the expectation values of the Pauli operators.
On any single qubit measurement, the value returned will either be +1 or −1. However,
upon looking at the same measurement over and over on N copies of the same state, the
expectation can be determined with Gaussian statistics and hence standard deviation given
√
by 1/ N. Note that the decomposition of ρ into the Pauli basis here is not a unique one.
There are in fact an infinite number of choices of bases which span the entire single-qubit state
space. However, it is often simpler to think about the decomposition in terms of the Pauli
measurements, which are easily attainable in many systems, especially the case for circuit
QED to be described later in this thesis. Specifically for a single qubit, only 3 observables
are necessary for complete state characterization, since the density matrix has 4 entries
along with the constraint tr(ρ) = 1. Therefore, to perform single-qubit state tomography
requires the ability to repeat measurements of 3 observables which determine the single-qubit
polarization.
Expanding the state tomography to two qubits, there are then 16 matrix elements of the
density matrix to be determined. Therefore, we find a set of 15 linearly independent operators
{M i } which span the two-qubit density matrix space. Then, ρ can be decomposed as

ρ = ∑ ci Mi , (2.36)
i
34 quantum information processing

where we can estimate {c i } from measurements of {M i }. Obtaining the expectation values

m i = tr(M i ρ), we then have

m i = ∑ tr(M i M j )c j . (2.37)
j

Analogous to the single-qubit case, one choice we can make for the {M i } is to use all of the
two-qubit Pauli operators, which are all pairwise combinations of the Pauli operators on each
qubit R ⊗ Q, where R, Q ∈ {I, X, Y , Z}. Then, the density matrix is given by

tr (R (1) ⊗ Q (2) ρ) (R(1) ⊗ Q (2) )

ρ= ∑ . (2.38)
R,Q∈{I,X,Y ,Z} 4

Therefore, the two-qubit quantum state tomography is now reduced to measuring two-qubit
correlation terms, such as XX, Y Y, ZZ, etc., in addition to single-qubit Pauli observables,
such as XI, IX, etc.
From the linearly independent measurements, ρ could be obtained through simply in-
verting tr(M i M j ). However, this method neglects the Hermiticity and positivity properties
which ρ must have. To account for this, we use the Cholesky decomposition to search for a
lower triangular matrix T which can be used to parametrize any Hermitian and positive-semi
definite matrix ρ as

T†T
ρ= . (2.39)
tr(T † T)

In the two qubit case, we have

⎛ t1 0 0 0⎞
⎜ ⎟
⎜ t5 + it6 0⎟
T =⎜ ⎟,
t2 0
⎜ ⎟ (2.40)
⎜ t11 + it12 t7 + it8 0⎟
⎜ t3 ⎟
⎝ t15 + it16 t13 + it14 t9 + it10 t4 ⎠

where the t i can be found from Maximum Likelihood Estimation (MLE) of the likelihood
function
16
L = ∑ α i (m i − tr(M i ρ)) ,
2
(2.41)
i=1

with α i as weighting factors depending on the sensitivity of certain measurements in the

 2.6. entanglement metrics 35

experiment [35]. We will go into more detail with respect to the merits and demerits of MLE
for two-qubit state determination in chapter 8.
Now given an experimentally determined density matrix ρ, we can try to quantify how
close it actually is to the ideal state we expected ∣ψ⟩. This performance metric is known as
the state fidelity F, and is given by

F = ⟨ψ∣ ρ ∣ψ⟩ , (2.42)

with values 0 ≤ F ≤ 1. The actual definition of fidelity varies throughout the literature,
√
sometimes actually given by ⟨ψ∣ ρ ∣ψ⟩ [36]. The distinction comes in as to whether the
quantity desired is a probability or a probability amplitude. Nonetheless, it is important to
note which of the definitions is used before comparing quoted values of the fidelity. As we will
see later in this thesis, the fidelity will be an important experimental metric for determining
the quality of states and we will be further discussing errors in its attainment in chapter 8.
Quantum state tomography becomes increasingly difficult with an increasing number of
qubits due to the increased state matrix space. Specifically, for a system of n qubits, the number
of measurements required to specify the states is 22n − 1. As a result, for systems of three
qubits or more, it can become prohibitively time-consuming to experimentally determine
the entire density matrix∗ . Instead, it may be favorable to obtain reduced information about
subsystems of the entire state, or to measure joint operators of multiple qubits, such as the
parity [37, 38], as opposed to the density matrix. Furthermore, as we will see in the next
section, entanglement metrics which are based upon a complete identification of ρ can be
difficult to compute, and other simpler experiments for entanglement quantification will
need to be developed.

2.6 Entanglement metrics

With quantum state tomography (section 2.5.2), we are able to completely reconstruct a quan-
tum state, whether it is pure or mixed and entangled or separable. As previously introduced,
a metric for quantifying the purity of experimentally produced states is P = tr(ρ2 ). Given a
d dimensional Hilbert space, we have the property 1/d ≤ P(ρ) ≤ 1. Another property of the
purity of a quantum state is that it remains invariant under unitary transformation, such as

∗
The current record is 8 qubits in a trapped-ion system requiring considerable computational effort [15]
36 quantum information processing

single-qubit and two-qubit operations. Experimentally, the purity can be a good indicator of
the decoherence present in the quantum system.
Although P and the fidelity to the targeted state F (2.42) give a considerable amount
of quantitative information about the quality of the states produced, we would like to have
further metrics which can quantify the degree of entanglement in the system. Note that this
entanglement which we wish to discuss will be bipartite entanglement, as a strict formalism
beyond two qubits is still an on-going topic of theoretical research [39–41].
To characterize and quantify entanglement, we introduce the concept of an entanglement
monotone E(ρ). Formally, it is defined as a functional that characterizes the strength of
genuinely quantum correlations with the following properties [42]

1. E(ρ) = 0 if ρ is a separable state.

2. E(ρ) = 1 if ρ describes a Bell state, which is maximally entangled.

3. E(ρ) is invariant under all local unitary operations.

4. E(ρ) cannot be increased by any combination of local operations with classical com-
munication channels operating on ρ.

Given any state ρ, E(ρ) will quantify the degree of entanglement between separable and
maximally entangled with a monotonic mapping. Entanglement monotones theoretically
only exist for bipartite entanglement[39], and so always refer to two-body density matrices.
Next, we discuss a relatively well-known entanglement monotone, known as the concurrence,
used as a metric across many quantum information experiments. Here, we will describe how
to calculate it given the case of pure or mixed states.

2.6.1 Concurrence
The concurrence is an example of an entanglement monotone for bipartite entanglement
characterization which is bounded between 0 and 1 [43]. Any pure two-qubit state ∣ψ⟩ can be
represented in terms of the computational basis states as

∣ψ⟩ = α00 ∣00⟩ + α01 ∣01⟩ + α10 ∣10⟩ + α11 ∣11⟩ . (2.43)

Then, the concurrence is defined as

C(ψ) = 2 ∣α00 α11 − α01 α10 ∣ . (2.44)

 2.6. entanglement metrics 37

1.0
Entanglement of formation
0.8

0.6

0.4
Entanglement of formation
Concurrence
0.2

0.0
0.0 0.2 0.4 0.6 0.8 1.0
Concurrence
Figure 2.10: Entanglement of formation versus concurrence. All entanglement monotones
for two qubits can be one-to-one mapped to each other. Here we see the relationship between
the concurrence and the entanglement of formation, two commonly quoted entanglement
monotones.

Another way of calculating the concurrence is from the matrix of correlations, or T

matrix, given by

⎛ XX XY XZ ⎞
⎜ ⎟
T = ⟨⎜Y X Y Y Y Z ⎟⟩ . (2.45)
⎜ ⎟
⎝ Z X ZY ZZ ⎠

It can be shown [39, 42] that

√
tr(T † T) − 1
C= . (2.46)
2
The concurrence is closely related with another entanglement monotone, known as the
entanglement of formation [43, 44], via the relation
1 √
E f (ρ) = h ( (1 + 1 − C(ρ))) , (2.47)
2
38 quantum information processing

where h(x) = −x log2 (x) − (1 − x) log2 (1 − x). Figure 2.10 shows the relationship between
concurrence and entanglement of formation from separable to maximally entangled states.
The concurrence is an especially interesting entanglement monotone because it can be
computed for mixed states as well. Given the full density matrix ρ, we first form the matrix
product R = ρ(σ y ⊗ σ y )ρ ∗ (σ y ⊗ σ y ). Taking the eigenvalues of R and arranging them in
decreasing order as {λ1 , λ2 , λ3 , λ4 }, the concurrence is then given by
√ √ √ √
C(ρ) ≡ max (0, λ1 − λ2 − λ3 − λ4 ) . (2.48)

Here, one of the drawbacks of C as an entanglement metric is the need to determine the full
quantum state ρ. Specifically as the number of qubits coupled together grows, determining
multipartite entanglement through determining the state becomes costly both in experimental
terms and computational requirements. Another caveat of needing to determine ρ is the
non-linear processing which is done, including maximum-likelihood estimation as well
as eigenvalue decomposition, resulting in convoluted error propagation. As a result, an
alternative method for quantifying entanglement can be sought, and that is to use witnesses,
which will be the subject of the next section.

2.6.2 Entanglement witnesses

Entanglement witnesses are observables which give expectation values that determine whether
a prepared state is entangled. It is formally defined as a Hermitian operator W, such that its
expectation value tr(ρW) > 0 for every separable state, but tr(ρW) < 0 for some entangled
states. This means that any negative expectation value guarantees entanglement, i.e. the
entanglement is ‘witnessed,’ whereas a positive value does not shed any information on the
state [39, 45].
We can visualize the full state space as in figure 2.11, with a central convex set of separa-
ble states, surrounded by concentric convex sets of increasing degree of entanglement. A
measurement of an entanglement witness is then given by a hyperplane slicing through the
space, which gives a value tr(Wρ) = c. All of the states which the hyperplane touches will
give that value. Therefore, a witnesses W can be thought of as parallel hyperplanes cutting
through the space, with different measured values c < 0 when the state is definitely not in the
separable set. When the hyperplane touches the separable state space or goes through it, then
c ≥ 0, indicating that the state is not necessarily entangled. Hyperplanes which are tangent
 2.6. entanglement metrics 39

separable
states

Figure 2.11: State space and entanglement witnesses. A convex set of separable states is sur-
rounded by concentric convex sets of increasing entanglement. The measurement of an en-
tanglement witness W is represented by a straight hyperplane which cuts through the space.
All states ρ along the hyperplane give a value tr[ρ] = c, where c < 0 for some entangled state.
Any witness which pierces through the separable state set will give c > 0. (Figure used with
permission from [45]. See Copyright Permissions.)

to the entangled states at the outer most edge of the entire space are optimal entanglement
witnesses to those states, as c takes the minimum possible value [41].
Perhaps the most well-known maximally entangled states are the Bell states,
1 1
∣Ψ± ⟩ = √ (∣0, 0⟩ ± ∣1, 1⟩) ∣Φ± ⟩ = √ (∣0, 1⟩ ± ∣1, 0⟩) . (2.49)
2 2
We can find a set of entanglement witnesses which, written in terms of two-qubit Pauli
operators as
1
WΨ± = (II ∓ XX ± Y Y − ZZ),
4 (2.50)
1
WΦ± = (II ∓ XX ∓ Y Y + ZZ),
4
would be optimal to these Bell states. Here, we can see that each Bell state has a unique
optimal witness which corresponds to it, and gives a minimum value of −1. These witnesses
demonstrate that in order to measure the expectation values tr(ρW), it is in fact not necessary
to have the full density matrix ρ, but just the expectation of some of the two-qubit Pauli
operators, XX, Y Y, and ZZ.
40 quantum information processing

These witnesses, although not entanglement monotones, can be used to place bounds on
measures such as the concurrence. In Ref. [45], it is shown that the quantity given by

B = −2 tr(ρW) (2.51)

is a lower bound on the concurrence of the system. Therefore, from a reduced set of measure-
ments of the quantum system, entanglement witnesses can be measured which quantitatively
restrict the degree of entanglement in the system. Furthermore, as most witnesses are simple
linear combinations of measurements, errors can be easily propagated, rather than forced
through layers of non-linear processing. These concepts of entanglement witnesses will be
applied to experimentally generated entangled states in chapter 8.

2.7 Bell tests

Traditionally, the idea of a Bell test is to devise an experiment which attempts to validate Bell’s
theorem that quantum mechanics is incompatible with local realism [46]. It was Bell who
showed that the presence of entanglement in quantum mechanics rules out the possibility
of pre-determined physical quantities prior to measurement. The test often involves the
violation of a Bell inequality, and finding a maximum value of correlation measurements
for distant objects. Here, coming from a different angle, we wish to extend the previous
section on entanglement witnesses and demonstrate that in fact a Bell test measurement need
not be applied as a validation of quantum mechanics but instead serve as another metric
of entanglement. From the point of view of quantum engineers, entanglement can be seen
as a resource, and having a high degree of entanglement will lead to a violation of a Bell
inequality.

2.7.1 Clauser-Horne-Shimony-Holt inequality

One of the more well-known Bell inequalities is the Clauser-Horne-Shimony-Holt (CHSH)
inequality [47]. It is often the inequality of choice in experimental violations as it has well-
defined classical and quantum mechanical bounds. To motivate this test, consider a source
of two particles which is capable of preparing the same set of two particles repeatedly. The
two particles are separated and sent to two distant parties, Alice and Bob.
Now suppose Alice and Bob can each perform two different projective measurements,
{M A , M A′ } and {M B , M B′ }, respectively (figure 2.12). Neither decides in advance which of
 2.7. bell tests 41

Bob
Alice
∣ψ⟩
⟨B⟩

⟨A⟩

source

Figure 2.12: Schematic for the CHSH test. Alice and Bob each receive one of a pair of particles
that have been prepared in an unknown state ∣ψ⟩. After performing various measurements
of the particles, they can compare answers and calculate the CHSH quantity given in (2.54).
For two classical or completely separable particles, i.e. ∣ψ⟩ = ∣ψ⟩A ∣ψ⟩B , there can be no set
of measurements which Alice and Bob can perform that would give a quantity larger than 2.
However, if the particles are initialized in an entangled state such as a Bell state 2.49, then for a
certain
√ choice of measurement angles, they can beat the bound of 2, reaching a maximal value
of 2 2.

the two measurements to perform but rather chooses randomly with a probability of 0.5 for
each. Upon performing a measurement simultaneously with Bob (Alice), Alice (Bob) obtains
either A (B) or A′ (B′ ), either of which can take on the outcomes +1 or −1.
Now let us form the quantity C = AB + A′ B + A′ B′ − AB′ and investigate its properties. C
is often referred to as the CHSH operator, and the measurements A, A′ , B, B′ can be thought
of as different axes onto which Alice and Bob can project their state. We can re-group the
terms of C into

AB + A′ B + A′ B′ − AB′ = B(A + A′ ) + B′ (A − A′ ), (2.52)

and since A, A′ = ±1, one of the two terms on the right hand side must be zero. As a result,
any single realization of measurements will necessarily give AB + A′ B + A′ B′ − AB′ = ±2. We
can take an expectation of the quantity, which must still be bounded,

E(C) = E(AB + A′ B + A′ B′ − AB′ ) ≤ 2. (2.53)

The expectation value of the C can then be distributed, and then we are left with the CHSH
inequality,

E(AB) + E(A′ B) + E(A′ B′ ) − E(AB′ ) ≤ 2, (2.54)

42 quantum information processing

where the terms on the left side are found by Alice and Bob repeating the experiment multiple
times and then classically multiplying their measurements.
However, if we let the particles that Alice and Bob share be quantum mechanical, now
they can be initialized as a Bell state,
1
∣ψ⟩ = √ (∣01⟩ − ∣10⟩) , (2.55)
2
before we separate the particles and send them to Alice and Bob for measurement. Now, as
a result of the two particles being in an entangled state, Alice and Bob will actually violate
the CHSH inequality (2.54) with an appropriate choice of measurements. Specifically, we
can use the Cartesian (or Pauli) basis with Alice measuring A → Z (1) , A′ → X (1) and Bob
√ √
measuring observables that are 45○ rotated, B → (−Z (2) − X (2) )/ 2, B′ → (Z (2) − X (2) )/ 2.
In this case, we then find expectation values
1 1 1 1
⟨AB⟩ = √ , ⟨A′ B⟩ = √ , ⟨A′ B′ ⟩ = √ , ⟨AB′ ⟩ = − √ . (2.56)
2 2 2 2
Placing these quantities into the left hand side quantity of (2.54), we then get
√
⟨AB⟩ + ⟨A′ B⟩ + ⟨A′ B′ ⟩ − ⟨AB′ ⟩ = 2 2. (2.57)

This is not a unique realization either, as other choices of measurements and other entangled
states can be used to violate the CHSH inequality (2.54) as well. However, the example given
√
above is the maximal violation and the value 2 2 is termed Cirelson’s bound [48].
Quantum mechanics thus violates the CHSH inequality (2.54). So what went wrong with
the classical derivation? We assumed that the values A, A′ , B, and B′ all existed indepen-
dently of measurement, suggesting pre-determined realism. We further assumed that Alice’s
measurement does not in any way affect Bob’s measurement, suggesting locality. Therefore,
violation of a Bell’s inequality supports the idea that nature is non-deterministic and non-local.
These aspects of Bell’s tests are the focus of numerous theoretical studies, looking to reconcile
quantum mechanics with non-local realism, and more recently, they are also the focus of
many experimental studies [49–51] looking to close certain loopholes in tests for ruling out
local hidden-variable theories, which we will not go into detail in this thesis. Instead, the
measurement of the CHSH operator can be thought of in terms of the entanglement in the
system.
 2.8. chapter summary 43

2.7.2 CHSH entanglement witness

In the interests of building a quantum information processor, we take a less controversial
point of view regarding the measurement and violation of a CHSH inequality. We can sweep
foundational issues of quantum mechanics under the rug momentarily, and look at the
violation of a Bell inequality not as a test of quantum mechanics, but as a measure of the
degree of entanglement in the system.
Although the violation of the CHSH inequality is traditionally found for a Bell state as a
45○ rotation of the measurements performed between Alice and Bob, we can instead let Alice
and Bob keep their measurements 90○ apart and perform a single-qubit rotation of 45○ on
the Bell state. So suppose we select the Cartesian axes such that C = XX − XZ + Z X + ZZ. In
fact, for this choice of measurements, the state that maximally violates the CHSH inequality
√
with a value of 2 2 is a single-qubit rotation of 45○ on one of the qubits in a Bell state, or
R y (−π/4) ∣Ψ+ ⟩. In this case C is actually an optimal entanglement witness [45] for this
(1)

specific state.
We can therefore see that the classical threshold for ⟨C⟩ = 2 is simply an offset value
of an entanglement witness: any measurement of ⟨C⟩ > 2 necessarily implies that the state
prepared is entangled and not separable; any measurement ⟨C⟩ < 2 just tells us that we
cannot comment on whether the state is separable or entangled. The maximal entanglement
√
attainable is signified by a measurement of ⟨C⟩ which approaches Cirelson’s bound of 2 2.
The CHSH operator is thus an extension of the entanglement witnesses discussed previ-
ously. Having the ability to measure two-qubit Pauli operators will permit the construction
of C and chapter 8 will demonstrate its measurement on a variety of generated separable and
entangled states.

2.8 Chapter summary

The previous discussions of this chapter have been general for any qubit implementation.
Building any simple quantum information processor will require single-qubit gates, an
entangling two-qubit gates, and a way for reading out the quantum state. For good single-
qubit control, we will want to have the ability to perform arbitrary rotations around the Bloch
sphere, perhaps combining rotations around the Cartesian axes x, y, and z. We have also now
√
seen how some two-qubit gates, such as the c-Phase and iSWAP, can arise from two-qubit
interaction Hamiltonians. These sets of gates can be a universal set for quantum computing
44 quantum information processing

and at the level of two-qubits, should permit the operation of some simple algorithms, such
as the Deutsch-Jozsa and four-level Grover’s search. Rounding out the quantum system with
a good quantum state measurement and we can be ready to develop a rudimentary two-qubit
quantum processor. Therefore, we will now leave the realm of general quantum computing,
and over the next few chapters motivate how we will bring some of these concepts to life in a
superconducting qubit architecture.
 CHAPTER 3

Superconducting Qubits and Circuit QED

P hysical implementations of qubits have taken many forms: nuclear spins, trapped-
ions, photons, and even electrical circuits. Yet, the operating principle of the qubit is
independent of its experimental formulation. The physics of the qubit i.e., of a simple two
level system, makes the quantum information processing described in the previous chapter
(chapter 2) possible. For an experimental realization, the challenge has been to find a pair of
quantum levels that can be addressed, coupled, protected from the environment, and scaled
up to a large number of qubits.
Achieving these often conflicting goals in circuit-based superconducting qubits has been
experimentally challenging. However, the potential of engineerable intrinsic qubit properties
and eventual mass-producibility based on a circuit design employing standard lithographic
fabrication techniques with all-electrical controls has driven continued progress. A particular
route for quantum computing with superconducting circuits has been to implement the
relatively new field of circuit quantum electrodynamics (QED) [30, 52–54], where quan-
tum optics is brought to a solid-state chip, coupling superconducting qubits to microwave
frequency photons.
This chapter will lay the foundation for superconducting circuit-based qubits and the
circuit QED architecture. It will serve as important background leading into chapter 4,
which will detail how circuit QED can be an excellent platform for quantum information

45
46 superconducting qubits & cqed

processing. First, section 3.1 will discuss the primary building blocks for the solid-state
quantum processor of this thesis, namely superconducting charge qubits. That is followed by
section 3.2 which will be an introduction to coupling multiple qubits. A review of basic cavity
quantum electrodynamics in section 3.3 will serve as a springboard for the concepts that will
be used in circuit QED (section 3.4), such as the strong and dispersive coupling regimes. We
end the chapter with a discussion about the relevant relaxation and decoherence properties
in circuit QED (section 3.5).

3.1 Superconducting qubits

Superconductivity provides an interesting foundation on which to study quantum effects. In

contrast to other solid-state implementations of qubits that aim to confine a small number
of microscopic quantum degrees of freedom, such as quantum dots, superconductors are
composed of a large number of paired electrons, or Cooper pairs, all of which have condensed
into a single ground state [55]. Quantum effects are then the result of macroscopic degrees of
freedom, and circuit elements comprised of these superconductors can be constructed with
tailorable interactions with other elements as well as the environment.
Ideally, non-dissipative circuit elements such as inductors and capacitors can be con-
structed with superconductors. However, combining these elements can only result in the
harmonic oscillator, which has evenly spaced energy levels, falling short of the two-level
addressability required for a qubit. Fortunately, superconductors also provide the only known
simultaneously non-linear and non-dissipative circuit element, known as the Josephson
junction, which will generate the necessary anharmonicity for artificial atoms. Artificial
atoms possess a rich non-uniformly spaced set of quantum mechanical levels, from which
two of them can be individually addressed for use as a qubit. The following sections will
review the non-linear Josephson junction and its application in the charge-based Cooper-pair
box and transmon qubits.

3.1.1 Josephson junction as a non-linear inductor

Physically, a Josephson junction consists of two superconducting electrodes separated by an
insulating oxide. Cooper pairs can tunnel coherently across the insulating barrier, with a
 3.1. superconducting qubits 47

supercurrent I that is given by

I = I0 sin ϕ(t) (3.1)

where I0 is the critical current (the maximum sustainable junction supercurrent), and ϕ(t)
is a time-dependent phase difference across the junction [55]. The phase difference evolves
in time in the presence of a potential V across the junction according to

dϕ
ħ = 2eV . (3.2)
dt
Now by taking the time-derivative of the supercurrent, we find what is commonly termed
the Josephson effect,

İ = (I0 cos ϕ)ϕ̇ (3.3a)

2eV I0
= cos ϕ. (3.3b)
ħ
Faraday’s law gives V = −L İ, which lets us identify the Josephson inductance as

Φ0
LJ = , (3.4)
2πI0 cos ϕ

where Φ0 = h/e is the magnetic flux quantum. This non-linear inductance combined with the
intrinsic capacitance of the Josephson junction, given by CJ , thus results in an anharmonic
oscillator which serves as the basis for a number of superconducting qubit topologies [56].

3.1.2 The Cooper-pair box qubit

One of the simplest Josephson junction based qubit designs is the Cooper-pair box (CPB)
[53, 57, 58]. This charge-based circuit is formed when a superconducting island is connected
to a reservoir of Cooper pairs through a junction. The Josephson effect permits the coherent
tunneling of the Cooper pairs between the island and the reservoir, while a gate voltage (Vg )
can also electrostatically modulate a Coulomb blockade tunneling effect. A standard CPB
geometry is shown in figure 3.1a.
Through circuit quantization [57, 59], the CPB Hamiltonian can be found to be

HCPB = 4EC (n̂ − ng )2 − EJ cos ϕ̂ (3.5)

48 superconducting qubits & cqed

(a) C

V

EJ EC

(b) Φ̃
() ()
EJ EC
C

V

() ()
EJ EC

Figure 3.1: The Cooper pair box (CPB). The standard CPB (a) consists of an island connected
to a superconducting reservoir through a tunnel junction and is capacitively coupled to a
electrostatic voltage bias. In the split CPB (b), the island is connected to the superconducting
(1) (2)
reservoir via two split junctions, with Josephson energies EJ and EJ . The superconducting
loop gives the ability to tune the effective EJ (3.8)b by threading an external magnetic flux Φ̃.

where n̂ is the integer-valued Cooper pair number operator, n  is the continuously variable
offset gate charge due to a dc bias, and ϕ̂ is the conjugate operator to n̂, representing the
Josephson phase. The first part of the Hamiltonian can be interpreted as the electrostatic
charging component with the relevant charging energy scale given by EC = e 2 /2C Σ , where
C Σ = Cg +CJ is the total capacitance to ground of the CPB. The second term of the Hamiltonian
reflects the energy across the non-linear inductor in the junction due to the Josephson effect,
with a scale given by the Josephson energy EJ ≡ I0 Φ0 /2π.
The CPB is more commonly designed with a pair of junctions in parallel (figure 3.1b),
forming a superconducting loop which allows the tunability of the tunneling (EJ ) portion of
the Hamiltonian. The split-pair of junctions forms a superconducting quantum interference
device (SQUID) such that an externally applied magnetic flux Φ̃ piercing the loop will control
the rate at which Cooper pairs tunnel in and out of the CPB. Now including the two junctions
with different Josephson energies EJ1 , EJ2 , we have a new Hamiltonian

H = EJ1 cos ϕ̂ − EJ2 cos (ϕ̂ − 2π Φ̃/Φ0 ) . (3.6)

 3.1. superconducting qubits 49

This can be rewritten as [53]

1/2
Φ̃ Φ̃
H = (EJ1 + EJ2 ) ∣cos (π )∣ (1 + d 2 tan2 (π )) cos(ϕ̂ − ϕ0 ), (3.7)
Φ0 Φ0

where d = (EJ1 − EJ2 )/(EJ1 + EJ2 ) reflects differences in the junctions, and ϕ0 is a phase offset
given by tan(ϕ0 + π Φ̃/Φ0 ) = d tan(π Φ̃/Φ0 ). For standard experimentally made junctions
(chapter 5) that are aimed to be identical, the junction asymmetry is typically d ∼ 0.1, small
enough to give the approximate flux-tunable CPB Hamiltonian:

HCPB = 4EC (n̂ − ng )2 − EJ (Φ̃) cos(ϕ̂ − ϕ0 ) (3.8a)

EJ (Φ̃) = EJmax cos(π Φ̃/Φ0 ). (3.8b)

We can explicitly write this Hamiltonian in the charge basis by using the relations

∂
n̂ = i (3.9a)
∂ϕ
n̂e i ϕ̂ = e i ϕ̂ (n̂ + 1) (3.9b)

to get

2 EJ
H = 4EC (n̂ − ng ) − ∑ (∣n⟩ ⟨n + 1∣ + ∣n + 1⟩ ⟨n∣) . (3.10)
2 n

The CPB can be operated as a charge qubit in the regime where EJ ≪ 4EC , such that the
Josephson coupling gives a small perturbation to lift the degeneracy at integer charge states.
By operating the CPB at a gate charge ng = ±0.5, the system can be reduced into a two-level
qubit system with a reduced Hamiltonian given by

EJ (Φ̃)
H ≈ 2EC (1 − ng )σz − σx , (3.11)
2
where we idenfity the standard spin 1/2 Pauli matrices σz → 2n̂ and σx → ∣n⟩ ⟨n + 1∣+∣n + 1⟩ ⟨n∣
in the two charge manifold. This Hamiltonian can be interpreted as a single spin in a magnetic
field given by B = EJ x̂ + 4EC (1 − ng )ẑ. Here, the eigenstates are superpositions of the charge
√
states with n = 0, 1, given by (∣0⟩ ± ∣1⟩)/ 2. Furthermore, another key aspect of the operating
point with ng = ±0.5 is the first-order insensitivity to fluctuations in ng , as can be seen in the
50 superconducting qubits & cqed

EJ /EC = 1 EJ /EC = 5 EJ /EC = 10 EJ /EC = 50

30
60
20
20
20
40
E j /EC

10 10 10 20

0 0 0 0
−1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1 −1 −0.5 0 0.5 1
ng ng ng ng
Figure 3.2: Charge dispersion. The energies of the lowest 5 levels of the charge qubit Hamil-
tonian (3.5), in units of the charging energy EC . For low EJ /EC ratio, we are in the Cooper
pair box regime, and the energies are parabolic functions of the offset charge ng , with avoided
crossings. Here, operation as a qubit is performed at charge ‘sweet spots’ ng = ±0.5 where the
energy levels are first-order insensitive to charge fluctuations. As the ratio of EJ /EC is increased
the levels become exponentially flatter, as we enter the transmon regime. Figure reproduced
from [61, 62].

dispersion diagram of figure 3.2. Performing experiments at this charge ‘sweet-spot’ is crucial
for obtaining longer coherence times [53, 60].

3.1.3 The transmon qubit

Starting with the charge-qubit Hamiltonian of (3.5), by operating in a different regime of the
ratio EJ /EC , it is possible to have a qubit that is optimized [61, 63] with respect to 1/ f charge
noise effects [64]. Specifically, when EJ ≫ EC , the CPB system switches to a system that is
best described as an anharmonic oscillator. In terms of physical realization, the transmon
modification involves the addition of a large shunting capacitance to increase the overall
capacitance to ground of the network, reducing EC (more transmon design details are given
later in this thesis in chapter 5).
To be more quantitative with regards to the dependence of the energy levels to EJ /EC , it is
useful to write out the full energy level expressions for (3.5). Switching over to the phase basis
[61, 62], analytical solutions can be expressed in terms of the special Mathieu’s functions
 3.1. superconducting qubits 51

meν (q, z) as,

1 −EJ ϕ
⟨ ϕ ∣ m ⟩ = √ exp[ing ϕ]me−2(ng +m) ( , ) (3.12)
2π 2EC 2

and the eigenenergies are then given by

E m (ng ) = EC a2(ng +k(m,ng )) (−EJ /2EC ), (3.13)

where a p (q) is Mathieu’s characteristic value and k(m, ng ) is an integer-valued function

which orders the eigenvalues. The effect of increasing EJ /EC can be seen in the level disper-
sion curves in figure 3.2. As the ratio increases, the levels flatten considerably and the ng
dependence of the first few levels disappears.
This flattening does not come for free, however, as another feature of the increasing EJ /EC
ratio is a reduced anharmonicity: the change in level spacing between adjacent transitions
decreases. Having sufficient anharmonicity is critical for operation as a qubit, since the case
of zero anharmonicity equates to a harmonic oscillator with levels that cannot be individually
addressed.
The Mathieu functions can be evaluated numerically through a truncated set of charge
basis states [62]. From these eigenenergies, we then define the charge dispersion,

є m = E m (ng = 0) − E m (ng = 1) (3.14)

for the mth energy band. In the CPB case with EJ < EC , є m ≈ 4EC . However, when EJ /EC ≫ 1,
an exponentially reduced charge dispersion can be found from the Mathieu solutions [61],
√ m 3
24m+5 2 EJ 2 + 4 −√8EJ /EC
m
є m ≃ (−1) EC ( ) e , (3.15)
m! π 2EC
which is illustrated in figure 3.2. In practice however, determining the dispersion from
the Mathieu solutions can become unwieldy and numerically intensive. Instead, we often
diagonalize the full charge qubit Hamiltonian using a truncation of up to ∼ 30 levels [62]. In
the limit of large EJ /EC , this treatment agrees very well with (3.15).
Now with regards to the anharmonicity, it is sufficient to use a perturbation to a harmonic
oscillator, expanding the cos ϕ in (3.5) to 1 − ϕ2 /2 + ϕ4 /24. The Hamiltonian then takes the
52 superconducting qubits & cqed

form of the Duffing oscillator

√ EC
H= 8EC EJ (b † b + 1/2) − EJ − (b † + b)4 , (3.16)
12
where b † and b are now creation and annihilation operators for simply the harmonic oscillator
portion of the cos ϕ expansion. Using perturbation theory and keeping only quartic terms of
the form (b † b)2 , the energy of the mth level can be found to be
√ 1 EC
E m ≃ −EJ + 8EJ EC (m + ) − (6m2 + 6m + 3) . (3.17)
2 12
The absolute anharmonicity between a transition between levels m + 1 and m and the next
lowest transition m and m − 1 is given by

α m = E m+1,m − E m,m−1 ≈= −EC , (3.18)

where E mn = E n − E m . Comparing this absolute anharmonicity to the ground to first excited

√
state transition of the transmon energy levels, E01 ≃ 8EJ EC , gives the relative anharmonicity,

r
αm = α m /E01 = −(8EJ /EC )−1/2 . (3.19)

This reflects an algebraic decrease in the anharmonicity with increasing EJ /EC . Although
as EJ /EC → ∞, the anharmonicity will be reduced α m r → 0, typical transmon performance

will be obtained without needing to reach this extreme. Since the charge dispersion reduces
exponentially with increasing EJ /EC , there is already sufficient band suppression before the
anharmonicity becomes small enough to make two-level addressability an issue. Henceforth,
we will interchange between notation in which we treat the transmon as a simple two-level
system, such that its Hamiltonian is just that for a simple spin 1/2,

ħ
Hq = ω q σz , (3.20)
2
and notation in which the full energy spectrum of the transmon is taken into account,

H q = ħ ∑ ω k ∣k⟩ ⟨k∣ , (3.21)

k

where ∣k⟩ are the exact Mathieu’s solutions from (3.12) and ω k = E k /ħ, with E k from (4.2.4).
The higher levels of the transmon will play a critical role in some of the interactions which
will be described later in this thesis.
 3.2. coupling superconducting qubits 53

3.2 Coupling superconducting qubits

Having introduced the charge-based superconducting qubit, we now move towards scaling
up the circuit, since for quantum information processing (chapter 2), coupling multiple qubits
is necessary. A circuit-based architecture makes this coupling an engineering challenge, and
just as fabrication procedures govern the relevant parameters in the single-qubit Hamiltonian
(3.5), they will also determine the strength and form of multi-qubit interactions. Furthermore,
analogous to the tunability of individual qubit parameters, a circuit-based approach will
permit dynamical electronic control to turn on and off interactions in-situ.
Given the circuit element nature of the qubits, the simplest way to couple them is to use
another lumped circuit element, such as either a capacitor or inductor. In this section, we
will discuss a few possible coupling schemes that have been suggested for charge qubits.

3.2.1 Fixed capacitive coupling

A mutual capacitance C m can be used as in figure 3.3a to couple two charge qubits, which have
Josephson energies EJ1 and EJ2 and charging energies EC1 , and EC2 . The resulting Hamiltonian
for this two-qubit device is given by [65]

⎛ E00 − 21 EJ1 − 21 EJ2 0 ⎞

⎜ 1 ⎟
⎜ − 2 EJ1 − 21 EJ2 ⎟
H=⎜ ⎟,
E10 0
⎜ 1 ⎟ (3.22)
⎜− 2 EJ2 − 21 EJ1 ⎟
⎜ 0 E01 ⎟
⎝ 0 − 2 EJ2
1
− 21 EJ1 E11 ⎠

where E n1n2 = EC1 (ng1 − n1 )2 + EC2 (ng2 − n2 )2 + E m (ng1 − n1 )(ng2 − n2 ). Here, n1 and n2 are
the excess Cooper pairs in the two CPBs, ng1 and ng2 are gate charges. The mutual coupling
energy term is given by E m = 4e 2 C m /(C Σ1 C Σ2 − C m
2 ). The four computational basis states are

for ∣n1 , n2 ⟩, n1 , n2 ∈ 0, 1.
This type of shared linear capacitance interaction can be used to perform a controlled
operation such as a cNOT (section 2.3.1). The diagonal elements of (3.22) point to the
presence of a gate-controlled ZZ-interaction. The off-diagonal Josephson energy terms result
in avoided crossings at the charge degeneracy points ng1 = ng2 = 0.5, split by EJ1 and EJ2
between the symmetric and anti-symmetric charge states ∣0⟩ ± ∣1⟩ of each qubit. A controlled
operation is possible through applying a gate pulse that would take ∣0, 0⟩ to ∣0, 1⟩, but would
54 superconducting qubits & cqed

(a) (b) Φ Φ
V V
C  C 

C  Cm C 
L
() () V () () () () V
EC EC EC EJ EC EJ
() ()
EJ EJ

Figure 3.3: Superconducting charge qubit coupling schemes. (a) Fixed capacitive coupling.
A mutual capacitance C m connects two CPB circuits, resulting in an always-on interaction.
(b) Tunable inductive coupling. Two split-CPB circuits are joined by a mutual inductance L,
which allows independently tunable loops via external magnetic fluxes Φ1 and Φ2 .

not be commensurate with the gate pulse frequency necessary to take ∣1, 0⟩ to ∣1, 1⟩ (More
details on actual operation can be found in [66]).
The direct capacitive coupling scheme has the topological advantage of requiring no
additional control lines for the two-qubit coupling. However, the capacitive interaction E m is
fixed and always on. Although the effective strength of the coupling is tunable via changing
the qubit frequencies, the gate charge modulation will necessarily move the qubits away from
their optimal charge gate bias points, resulting in significant coherence time degradation.
Furthermore, this scheme is limited in its scope and error performance as the number of
qubits scales up since it only couples nearest neighbors, making operations between far apart
qubits in a chain of multiple qubits (figure 3.4a) costly in terms of resources.
Nonetheless, such fixed capacitive coupling has resulted in the first superconducting
qubit coherent dynamics experiments [65], as well as the first demonstration of a cNOT in
a solid-state system [66]. Similarly, fixed capacitive coupling has been implemented with
Josephson phase qubits, with the generation and state tomography of entangled states [67].

3.2.2 Tunable inductive coupling

Another choice for coupling is to share a common inductance between multiple Josephson
charge qubits. The shared inductor L combined with the capacitance of the charge qubits will
form an LC-oscillator mode for the coupling. Such a scheme [68, 69] lends itself towards
a more scaleable architecture, as it is not limited to just nearest-neighbor qubit coupling.
Figure 3.3b shows such a circuit where now the interbit coupling strength is flux-controllable.
 3.2. coupling superconducting qubits 55

By tuning the gate voltage and threaded flux of each CPB, it is possible to turn on either a
single-qubit or a two-qubit interaction regime. By setting the flux through the second qubit
(one on right hand side) loop to Φ2 = Φ0 /2 and the gate voltage to V2 = (2ng2 + 1)e/Cg2 , qubit
1 (one on left hand side) in the lowest two charge states ∣0⟩ and ∣1⟩ is individually addressable
with a Hamiltonian given by
1
H = EC1 (1 − ng1 )σz(1) − EJ1 (Φ1 , Φ̃, L)σx(1) (3.23)
2
where ng1 = Cg1 Vg1 /e and the Josephson energy scale is tunable by both the external flux
through the common inductance Φ̃ as well as the local flux Φ1 . At any flux bias which is not
Φ0 /2 in both loops there is a separate two-qubit interaction which is due to the persistent
current I = I1 + I2 circulating through the common inductance. The two-qubit coupling
Hamiltonian is then given by
1
H int = L(I1 + I2 )2 , (3.24)
2
where the current through each CPB loop is given by

Φ1 + Φ2 + L(I1 + I2 )
I1(2) = 2I c1(2) cos ϕ1(2) sin (π ) (3.25)
Φ0
In the first two charge level basis, this can be simplified into an XX interaction term, which
is similar to a ZZ interaction (section 2.3.2), but with a re-labeling of states.
By tuning the gate charge such that the first two charge levels are degenerate at ng1 = ng2 =
1/2 for each qubit, the σz terms can be turned off, and the reduced Ising-like Hamiltonian of
the system is then

H = −EJ1 σx1 − EJ2 σx2 + Πσx1 σx2 , (3.26)

where Π encapsulates the coupling (details given in [69]). The eigenstates of this Hamiltonian
√
are ∣+, +⟩ , ∣+, −⟩ , ∣−, +⟩ , ∣−, −⟩ where ∣±⟩ = (∣0⟩ ∓ ∣1⟩)/ 2, representing having rotated to
the basis of the XX interaction. In this four level manifold, it is then possible to produce a
conditional phase gate by tuning all of the energies to be the same, EJ1 = EJ2 = Π = −πħ/4τ
for a fixed amount of time τ, where the two-bit states ∣+, +⟩ , ∣+, −⟩ , ∣−, +⟩ are left the same,
but ∣−, −⟩ → − ∣−, −⟩.
It is important to note the fundamental difference of this scheme having a switchable
coupling (without needing to move the qubit frequencies) as opposed to the effective tunable
56 superconducting qubits & cqed

coupling in the capacitive case (via detuning the qubits from the interaction point). Similar
mutual inductance schemes with tunable coupling have been implemented in flux qubits,
where the shared inductance is further enhanced with the addition of a Josephson junction in
series [70–72]. In such systems, the coupling occurs through a magnetic-dipole interaction
and can be relatively stronger than in charge qubits.

3.2.3 Quantum bus coupling

The previously discussed qubit coupling schemes deal with using explicit lumped elements in
the circuits, using either a capacitor or an inductor. Experimentally controlling fixed coupling
designs such as the capacitor have been challenging due to electrostatic cross-talk between
different parts of the coupled circuit, while the inductive coupling can result in many mutual
qubit couplings requiring more complicated pulsing schemes for turning on and off specific
interactions. The coupling via the shared inductor circuit as discussed in the previous section
is actually an interesting direction, as it uses the idea of coupling to an electromagnetic mode
of the induced LC-oscillator. One can take this idea a step further to achieve the coupling
through the distribution of quantum information over an entirely separate quantum degree
of freedom. In such a scheme, multiple qubits would be coupled to a shared quantum bus, in
analogy to classical bits of information transmitted along a data bus in a classical processor.
Multi-quanta interactions can then be non-local, permitting operations over an arbitrary
pair of qubits.
A bus coupling differs from the linear, or nearest-neighbor arrangement of qubits (fig-
ure 3.4a), in which performing a two-qubit interaction between non-connected qubits requires
multiple pair-wise operations. This direct coupling can result in a rapid build-up of errors due
to the large number of gates required to communicate between distant qubits, or by analogy
to classical computers, an increased number of clock cycles per operation. The bus coupling
also differs from a full mutually coupled network of qubits, as depicted in figure 3.4b. In such
a scheme, every single qubit is coupled to every other qubit, forming a matrix of interactions.
This makes the ability to simply address a single qubit difficult, and requires a switchable
coupling to turn on and off interactions between qubits.
So how could one physically realize a quantum bus? There has been pioneering work
with a quantum bus coupling of trapped-ion qubits, where the quantized motion of the ions
as phonons serve as the bus. The phonon quantum bus has led to the ability to perform
universal quantum operations as well as quantum algorithms in ion qubits [73]. Another
 3.2. coupling superconducting qubits 57

(a)

(b)

Figure 3.4: Charge-qubit coupling networks. (a) As the coupling between two charge qubits
can be achieved through a discrete lumped element, such as a capacitor or an inductor, the
simplest scheme for scaling up to more elements is to chain up more discrete lumped elements
between each charge qubit. (b) One of the situations which can arise from attempting to couple
charge qubits on a circuit with capacitances is the possibility of mutual couplings between all
pairs of charge qubits. This makes the network of coupled charge qubits very large, and to
address only a single qubit can become quite difficult.
58 superconducting qubits & cqed

Figure 3.5: Quantum bus coupling cartoon. A quantum bus coupling attempts to use a separate
quantum degree of freedom. Schematically, it has its analog in a classical instruction bus, with
multiple bits locked into the same bus. Here, we can imagine the capacitive coupling of multiple
charge qubits to a some feedline which will contain the quantum bus.

natural candidate for carrying quantum information is the photon. Photons can be highly
coherent and interact with objects over distances greater than their wavelengths. To have
increased interaction strength with a photon bus, we can employ the techniques of cavity
quantum electrodynamics (QED) [74, 75], in which a single atom is coupled to a single cavity
mode.
For the purposes of quantum computing with a quantum bus in superconducting qubits,
cavity QED [74, 75] has been adapted into circuit quantum electrodynamics [30, 52, 53],
where the photon bus is realized as a microwave frequency on-chip resonator and the atoms
are replaced with superconducting qubits, such as Cooper pair boxes or transmons. It is
with this architecture that we have realized a full two-qubit solid-state quantum processor.
However, to motivate the quantum bus coupling in circuit QED, it is thus important to first
review the key aspects of atomic cavity QED.

3.3 Cavity quantum electrodynamics

In cavity QED, individual atoms are passed through a Fabry-Perot cavity and interact co-
herently with the harmonic oscillator excitations, which are optical [76] or microwave [77]
photons. Figure 3.6 illustrates the atom-photon field interaction. The full coupled photon-
atom system is described by the Jaynes-Cummings (JC) Hamiltonian

1 ħω a
H = ħωC (a † a + ) + σz + ħ (a † σ− + aσ+ ) (3.27)
2 2
 3.3. cavity quantum electrodynamics 59

g κ

γ
ttransit
Figure 3.6: Illustration of cavity QED. A two-level atom passes through a Fabry-Perot cavity
over a transit time t, during which the atom undergoes a coherent interaction with photons
contained in the cavity with a strength . Photons can leave the cavity at a rate κ and the atom
decays via non-cavity modes at a rate γ.

where the first term corresponds to photons with excitation ħωC comprising the electro-
magnetic energy of the cavity, the second term represents the individual spin-1/2 atom with
transition energy ω a , and the third term represents a dipole interaction between the cavity
and the atom within the rotating wave approximation (RWA). The interaction term, com-
monly known as the vacuum Rabi coupling, is the result of the quantization of the electric
dipole coupling, and corresponds to coherent absorption (σ+ a)/ emission (σ− a † ) of a photon
from/to the electromagnetic field at a rate .
Although the Jaynes-Cummings Hamiltonian only describes a general two-body interac-
tion between an atom and photon-field, a real quantum system inevitably couples to objects
in the classical environment. Some of these incoherent processes in the cavity QED system
include photon leakage and absorption, given by a rate κ which is often encapsulated by
the transparency of the mirrors. This photon decay is actually paramount for probing the
system, as photons which enter and transmit through the cavity reveal the internal dynamics
of the system. The atom can also be subject to decay, either through a radiative decay via
a coherent interaction with the cavity photons, or through interaction with modes outside
of the Jaynes-Cummings realm. We can denote the decay of the atom due to all non-cavity
channels as γ.
Depending on the values of the atom and photon energies, there can be different signatures
of the interaction in cavity QED. Understanding these regimes will be critical to successfully
operate the quantum bus for quantum information processing.
60 superconducting qubits & cqed

3.3.1 Strong coupling regime

The atom and cavity are in what is termed the strong coupling regime when the interaction
 is much stronger than the atom and cavity decay rates, γ and κ, respectively. When the
photons in the cavity and atom are in resonance with each other (ωC = ω a ), the interaction
fully hybridizes the energy levels of the combined atom and photon field system, resulting in
dressed-state eigenstates in the one-excitation manifold as follows
1
∣+⟩ = √ (∣↑, 0⟩ + ∣↓, 1⟩) (3.28a)
2
1
∣−⟩ = √ (∣↑, 0⟩ − ∣↓, 1⟩) . (3.28b)
2
These are simply the symmetric and anti-symmetric combinations of a single excitation
in either the atom or the cavity. Here, the atom and photon can freely exchange a single
quanta at a rate  as a vacuum Rabi oscillation. Later in this chapter, such a direct cavity
swap interaction will be an important mechanism for inducing coupling between multiple
quantum degrees of freedom.
In the strong coupling regime, more generally, there can be any number of excitations,
resulting in a ladder of states with eigenstates in the n-excitation manifold given by
1
∣+⟩n = √ (∣↑, n − 1⟩ + ∣↓, n⟩) (3.29a)
2
1
∣−⟩n = √ (∣↑, n − 1⟩ − ∣↓, n⟩) (3.29b)
2
√
The energies of these states are split by 2 n and demonstrate a built-in anharmonicity
which can allow the strong-coupling regime of cavity QED to behave as a multi-level qubit.

3.3.2 Dispersive coupling regime

The cavity QED system can also be operated in a dispersive regime in which the atom and
cavity do not directly exchange energy. This is achieved through detuning the atom from the
cavity such that ∆ = ω a − ωC ≫ . Given this condition, this regime can be studied using a
second-order perturbative expansion in /∆, to give the dispersive JC Hamiltonian,

2 1 ħω a
H ≈ ħ [ωC + σz ] (a † a + ) + σz . (3.30)
∆ 2 2
 3.4. circuit qed 61

This new Hamiltonian reflects a re-diagonaliation of the full JC Hamiltonian given the
dispersive condition. To second order, the eigenstates of this dispersive Hamiltonian coincide
with those of the full Hamiltonian. The interaction is now transferred into an atom state
dependent shift of the harmonic oscillator frequency (the first term), which can now take
either of two values, ω′C = ωC ± 2 /∆. The dispersive shift plays the central role for atom state
interrogation in this regime via a quantum non-demolition (QND) measurement and will be
the basis for multiple qubit readout in the framework of circuit QED to be discussed later.
Another way of interpreting the interaction is to re-order the terms in the Hamiltonian
of (3.30) to be
1 ħ 22 † 2
H ≈ ħωC (a † a + ) + (ω a + a a + ) σz . (3.31)
2 2 ∆ ∆
With this arrangement, the interaction has now been moved to the right most term, behaving
as a shift of the atom transition frequency. Specifically, the first term 22 /∆n, where n = a † a,
reflects a photon number-dependent Stark shift while the 2 /∆ term is a Lamb shift due to the
electromagnetic vacuum [30, 78]. The Stark shift is a critical feature of the dispersive regime
as it allows for an effective means to tune the atom transition frequency with microwave
pulses, which will be discussed in more detail in regards to multi-qunta interactions for
circuit QED later in this thesis (section 3.4.2).

3.4 Circuit QED

With some of the basic concepts of cavity QED under our belt, we can now move on to its
analog with superconducting circuits and develop the framework for its use as a quantum
bus architecture. Specifically, this section will deal with circuit QED using transmon charge
qubits section 3.1.3 and we will revisit the strong and dispersive coupling regimes.

3.4.1 Coupling a transmon to a coplanar waveguide resonator

We can translate the cartoon of cavity QED (figure 3.6) into a circuit geometry as shown in
figure 3.7. The Fabry-Perot cavity is now replaced by a microwave-frequency co-planar wave-
guide (CPW) transmission line resonator. The CPWs can be made out of superconducting
material which can be low loss below their critical superconducting temperatures, allowing
for high quality factor (Q) resonators. The gaps in the center-pin transmission line serve
as the mirrors of the cavity, with microwave radiation forming standing waves within the
 62 superconducting qubits & cqed

Figure 3.7: Illustration of circuit QED. A two-level atom passes through a Fabry-Perot cavity
over a transit time t, during which the atom undergoes a coherent interaction with photons
contained in the cavity with a strength . Photons can leave the cavity at a rate κ and the atom
decays via non-cavity modes at a rate γ.

center-stripline. Through careful engineering of these capacitive gaps and the length of the
center stripline, the resonant frequency and quality factor of the resonators can be designed
(chapter 5).
Following Ref. [62], the transmission line resonator circuit can be quantized. For a
transmission line of length d, capacitance per unit length c, and inductance per unit length l,
the Hamiltonian can be expressed as
1
H = ħ ∑ ω n (a †n a n + ) (3.32)
n 2
√
with resonant frequencies ω n = nπ/d l c. For the purposes of the experiments discussed
in this thesis, we will be working in the vicinity of just the first mode, with n = 1, and as a
√
result we will write the cavity Hamiltonian with frequency ωC = π/d l c, without the sum
and subscripts.
The coupling between a transmon qubit and the transmission line resonator is an elec-
trostatic capacitive interaction∗ . We can place the transmon near either end of the CPW to
couple to a voltage antinode for the n = 1 mode (λ/2) of the resonator. The Hamiltonian for
this combined system will be the sum of the transmon Hamiltonian (3.21), the transmission
line resonator Hamiltonian (3.32) and a dipole interaction term from the product of the

∗
This is due to the physical size of the transmon being much smaller than the wavelength of the resonator,
allowing a lumped element interpretation.
 3.4. circuit qed 63

C
CJ
V Φ
EJ
CB

Figure 3.8: Reduced transmon coupling to CPW schematic. The transmon is coupled to the
CPW transmon line via a gate capacitance C  . A split pair of junctions with EJ and EC are in
parallel with a shunt capacitance C B .

voltage in the cavity, V0 (a + a † ), where V0 is the zero point root mean-squared voltage, with
the charge of the transmon, 2ne,
2
H = 4EC (n − ng ) − EJ cos ϕ + ħωC a† a + 2βneV0 (a † + a). (3.33)

Here, β is a voltage division ratio, defined by the ratio of the gate capacitance to the total
capacitance. We can view the reduced capacitance network of the transmon in a CPW as
shown in figure 3.8. The entire transmon coupled CPW circuit is presented in detail in
chapter 5. We can express the values of the parameters of (3.33) in terms of the reduced
network as,
e2
EC = (3.34a)
2C
√Σ
ħωC
V0 = (3.34b)
cL
C
β= (3.34c)
CΣ
C
CΣ = , (3.34d)
CJ + C  + C B

where C B is the shunt capacitance, C  is the gate capacitance, c is the capacitance per unit
length of the resonator, and L is the length of the resonator. We can write the Hamiltonian in
64 superconducting qubits & cqed

the basis of the eigenstates of the transmon ∣ j⟩ as,

H = ħωC a† a + ħ ∑ ω j ∣ j⟩ ⟨ j∣ + ħ ∑ i, j ∣i⟩ ⟨ j∣ (a + a † ) , (3.35)

j i, j

where ħi, j = 2eV0 β ⟨i∣ n ∣ j⟩ are dipole coupling energies which involve many charge states,
as the matrix elements for different transitions will all explicitly contribute. In the asymptotic
large EJ /EC limit, the dipole coupling is given by

√ eV0 EJ 1/4
i j ≈ 2 β( ) ⟨i∣ (c − c † ) ∣ j⟩ (3.36)
ħ 8EC
where c and c † are lowering and raising operators for the transmon energy levels. For nearest-
neighbor energy levels the coupling is given by

eV0 √ EJ 1/4
 j, j+1 = β ( 2( j + 1) ( ) ), (3.37)
ħ 8EC

and is the dominant contribution to the coupling in the large EJ /EC limit [61].
When there is sufficient anharmonicity such that the transmon can be operated as a qubit,
we can keep just the first two levels and use the Pauli spin operator notation:

ħωq
H = ħωC a† a + σz + σx (a + a † ) . (3.38)
2
By making the rotating wave approximation (RWA), given that ωC ≃ ωq and ωC ≫ , counter-
rotating terms, a† σ+ and aσ− can be neglected (where σ± = (σx ± iσ y )/2), so that then we
recover the Jaynes-Cummings Hamiltonian as discussed in section 3.3,

ħ
H = ħωC a† a + ωq σz + ħ (aσ+ + a† σ− ) . (3.39)
2
More generally for the multi-level transmon, the Hamiltonian is given by

H = ħωC a† a + ħ ∑ ω j ∣ j⟩ ⟨ j∣ + ħ ∑  j, j+1 (∣ j + 1⟩ ⟨ j∣ a + h.c.) . (3.40)

j j

Similar to cavity QED, we will be able to access a dispersive regime in circuit QED, described
in the following section.
 3.4. circuit qed 65

3.4.2 Dispersive regime of circuit QED

We have seen now that the transmon qubit can be in a strong coupling regime with a trans-
mission line cavity. By tuning the energy levels of the transmon away from resonance with
the cavity, we enter a dispersive regime (section 3.3.2) when  j, j+1 ≪ ω j+1, j − ωC . A unitary
transformation, H → U HU † , similar to the one used for the simple atom case can be made,
with
⎡ ⎤
⎢ ⎥
U = exp ⎢⎢∑ λ j ∣ j + 1⟩ ⟨ j∣ a − h.c.⎥⎥ (3.41)
⎢ j ⎥
⎣ ⎦
and expanding to second order in the small parameter λ j =  j, j+1 /(ω j, j+1 − ωC ). We are then
left with a diagonalzed Hamiltonian,

H/ħ = ∑ ω j ∣ j⟩ ⟨ j∣ + ωC a † a + ∑ χ j, j+1 ∣ j + 1⟩ ⟨ j + 1∣ − χ01 a † a ∣0⟩ ⟨0∣

j j

+ ∑ (χ j−1, j − χ j, j+1 ) a a ∣ j⟩ ⟨ j∣
†
(3.42a)
j=1

with dispersive couplings χ i j given by

i2j
χi j = . (3.43)
ω i j − ωC

Taking a two-level approximation for using the transmon as a qubit, the dispersive Hamilton-
ian then takes the form
ħ ′
H= ω σz + ħ(ω′C + χσz )a † a, (3.44)
2 q
where the qubit transition frequency ω′q = ω01 + χ01 and the cavity frequency ω′C = ωC − χ12 /2
are both Lamb-shifted. The dressed transitions here give a dispersive Hamiltonian similar to
the one from traditional cavity QED (3.30) but with a different transmon state dependent
shift given by
χ12
χ = χ01 − . (3.45)
2
Using the asymptotic expression for i j , the shift can be approximated as

2 EC
χ≈− , (3.46)
∆ ħ∆ − EC
66 superconducting qubits & cqed

where ∆ = ω01 − ωC .

3.4.3 Strong dispersive regime

The strong dispersive regime refers to the situation when the dispersive shift χ (3.45) is
greater than the linewidths of the qubit, γ, and of the cavity, κ. Figure 3.9 shows both the
representative spectra of the cavity and of a single qubit system (using simplified two-level
transmon approximation) signifying the strong dispersive limit. With χ > γ, the spectrum of
the qubit can be resolved into individual photon number peaks. This qubit number-splitting
pattern theoretically allows for the determination of the mean photon number state of the
cavity. The qubit frequency peaks will be located at ω n = ωq + 2n χ, and multiple peaks will be
observed given a thermal or coherent distribution of photons in the cavity. For a transmon,
these photon number dependent shifts will be slightly different due to the non-linearity
introduced into the cavity by the relatively smaller anharmonicity [79, 80]. Nonetheless, the
qubit spectrum can therefore be used to measure the photon number in the cavity.
With the cavity shift greater than the cavity linewidth, the transmission through the cavity
becomes a non-linear function of the state of the qubit. The cavity transmission is Lorentzian,
centered around frequency ωC + χ with the qubit in the ground state and centered around
frequency ωC − χ when in the excited state. Here, the state-dependent transmission can be
written as
κ/2
T= . (3.47)
ω − (ωC ± χσz ) + iκ/2

Now, when χ > κ, the transmission of an applied drive at ωC + χ will be a nonlinear function
of the qubit state, which permits a projective QND readout very much in the way a Stern-
Gerlach experiment can distinguish a spin polarization. This will be discussed in more detail
with respect to the joint readout of a multi-qubit state in chapter 4.

3.5 Qubit decoherence

As the transmon qubits are electromagnetic circuits, there are a number of factors in their
environment which can degrade their performance as quantum degrees of freedom. All qubit
errors can be classified as either relaxation and dephasing.
 3.5. qubit decoherence 67

1.0 (a)
Transmission

∣⟩ ∣⟩
0.0
6.97 6.98 6.99 7.00 7.01 7.02 7.03
Frequency (GHz)
0.25 (b) n=
Reduction of Amplitude

n=
e −n̄ n̄ n
0.20 P(n) = n! n=
n̄ = 
0.15 n=

0.10 n=

0.05 n=

0.00
5.86 5.90 5.94 5.98 6.02
Frequency (GHz)
Figure 3.9: Strong dispersive regime of circuit QED. (a) Theoretical state dependent cavity
transmission reveals two Lorentzian peaks for the case of ∣0⟩ and ∣1⟩, separated by 2χ (Here
assuming χ/π = 20 MHz). (b) Qubit spectroscopy shows multiple peaks corresponding to
different cavity-photon number. The weighting of these peaks is given by a simple Poisson
distribution as a function of the mean number of photons in the cavity n̄ = 2.
68 superconducting qubits & cqed

3.5.1 Relaxation and the Purcell effect

Relaxation and heating, are processes which result in the de-excitation and excitation of the
qubit, respectively. Fluctuations in some environmental parameter M (such as the flux Φ
or voltage V ), can be treated as a perturbation to the qubit Hamiltonian as H̃ = ξ̂M, where
ξ̂ is an operator which couples to the qubit eigenstates. The noise that specifically results
in relaxation and heating of the qubit occur at the qubit frequency ωq . By using first-order
time-dependent perturbation theory (Fermi’s golden rule), transition rates between the levels
can be found to be [81]
1 2
Γ↑↓ = 2
∣⟨∣ ξ̂ ∣e⟩∣ S M (∓ωq ), (3.48)
ħ
where S M (∓ωq ) reflects the spectral density of the noise associated with environmental para-
meter M at the frequency of the qubit. The relaxation time is given by T1 = 1/(Γ↑ + Γ↓ ). Here,
the qubit acts as a spectrometer of noise at its transition frequency [81]. Note also that the
qubit will distinguish between a positive frequency component of noise, corresponding to re-
laxation of the qubit and excitation of the environment, and a negative frequency component,
corresponding to excitation of the qubit and relaxation of the environment. To understand
the qubit relaxation process thus requires investigating the environmental sources of noise at
the qubit transition frequency and determining how strongly they couple to the qubit.
For a charge-based qubit, an intrinsic relaxation rate can be associated with a number of
noise processes, including flux fluctuations, quasiparticle tunneling, and dipolar radiation.
These are dealt with in detail for Cooper pair box qubits in Ref. [53] and for transmon qubits
in Ref. [61]. However, the estimated relaxation rates due to these factors are small compared to
the relaxation rate due to voltage coupling to the electromagnetic environment. Particularly,
in the circuit QED setup, in which the environmental impedance is well-controlled, the qubit
relaxation rate can be completely understood up to an intrinsic limit suggestive of a lossy
dielectric [82].
Starting with a two-level truncation to the transmon qubit, let us formally treat the
relaxation due to voltage noise. By capacitively coupling the qubit to an environmental
impedance Z(ω), voltage fluctuations δv on this line couple via the charge degree of freedom
n̂. Here, the voltage noise coupling operator is

ξ̂ = 2eβδV n̂ (3.49)
 3.5. qubit decoherence 69

where β is the voltage division ratio, given by C  /C Σ . With an environmental impedance

Z(ω), the zero-temperature voltage spectral density can be found from the Johnson-Nyquist
formula,

SV (ω) = 2ħωRe[Z(ω)]. (3.50)

Then, from Fermi’s golden rule (3.48), the effective decay rate Γ due to voltage coupling is
then determined to be
Re[Z(ωq )]
Γ = 16πe 2 β 2 ωq ∣ ⟨0∣ n̂ ∣1⟩ ∣2 . (3.51)
h
For the transmon qubit, the charge matrix element between the first two levels (3.37) is [61]

1 EJ 1/4
⟨0∣ n̂ ∣1⟩ = √ ( ) . (3.52)
2 8EC
√
Then, by using the expressions EC = e 2 /2C Σ , ωq = 8EJ EC , Γ can be simplified to

(ωq C  )2 Re[Z(ω)]
Γ= . (3.53)
CΣ
Therefore, the voltage noise effect on the relaxation rate occurs through the form of the
dissipative environmental impedance Re[Z(ω)].
Interestingly enough, the above result can be found using a simple circuit model as
well [82]. For the capacitive coupling circuit shown in figure 3.10, we can combine the gate
capacitance C  with the environmental impedance Z(ω) into an effective resistor given by
R = 1/Re[Y(ω)] where the admittance Y(ω) is

ω2 C 2 Z(ω) + iωC 
Y(ω) = . (3.54)
1 + ω2 C 2 Z 2 (ω)

The relaxation rate is then simply found from the decay time 1/RCeff , where Ceff = C Σ − C  ,
giving the effective capacitance of the transmon without the effect of the gate. Then, we find
the same result as in (3.53) with this treatment.
This formalism is very powerful, as it tells us that by identifying the environmental
impedance Z(ω), we are able to predict the relaxation rates of the qubit. Specifically in the
case of a qubit placed within a single-mode cavity, this reproduces the well-known Purcell
70 superconducting qubits & cqed

C

Z(ω)
C

Figure 3.10: Environmental coupling of transmon. The voltage relaxation rate of the transmon
qubit can be found from a simple RC decay time. The transmon capacitance C is coupled to an
environmental impedance of Z(ω) through a coupling capacitor C  .

[83] result,
2 κ
γκ = . (3.55)
(ωC − ωq )2

The Purcell effect reflects the altered spontaneous emission rate due to coupling to cavity
photons which decay at a rate κ. For the case of circuit QED, this simplified single-mode
Purcell treatment is unfortunately inadequate as a result of strong coupling to higher modes
of the transmission line cavity. Heuristically, this strong dependence on higher modes can be
√
seen from the increasing coupling strength with mode number n =  n + 1 as well as the
increasing decay rate of the nth harmonic, κ n = (n + 1)2 κ. The multi-mode cavity effects on
the Jaynes-Cummings physics is still a topic of ongoing theoretical research. However, it is
possible to still predict the relaxation rates due to the multi-mode Purcell effect by using the
circuit formalism and considering the coupling of the qubit to a distributed transmission-line
resonator. These calculations have been shown to agree strikingly well with experimental
predictions, up to a global lossy limit with a Q ∼ 50, 000 − 70, 000, as described in Ref. [82].
For the purposes of this thesis, the investigations of Ref. [82] demonstrate that our current T1
is modeled and well-understood, and improvements are the subject of current research.

3.5.2 Dephasing
Dephasing is generally understood as the fluctuations of the qubit transition frequency due
to coupling to the environment. Low frequency noise far below the transition frequency can
cause the qubit to accumulate a random phase. Qubit relaxation will dephase the qubit at a
 3.5. qubit decoherence 71

rate Γ↓ /2, which can be shown from a Bloch equation treatment [62]. In addition, there can
be fluctuations to the transition frequency which occur over the course of a decay lifetime,
labeled with a dephasing rate of Γϕ . We can call the total dephasing the sum of these rates
Γ2 = Γ↓ /2 + Γϕ . In the case that Γϕ = 0, the total dephasing rate is given by Γ↓ /2, such that the
dephasing time T2 = 2T1 .
Dephasing can be interpreted as the decay of off-diagonal density matrix elements. In a
two-level system, given a noise power S M (ω), this exponential decay is given by [61, 84],

1 ∞ dω sin2 (ωt/2)
ρ01 (t) = exp (− ∫ S M (ω) ). (3.56)
2 −∞ 2π (ω/2)2

A specific noise spectrum that can contribute to dephasing is 1/ f noise in the parameters
that determine the qubit transition frequency ω01 . The 1/ f power spectrum is given by

2πA2
S M (ω) = (3.57)
∣ω∣

and is a typical noise spectrum for charge, flux, and critical current, all of which can vary
ω01 . The parameter A determines the overall amplitude of the fluctuations and have been
measured in various separate experiments [85–88]. In the case of 1/ f noise, the limits of the
integral in (3.56) are between fmin and ω01 /2π, where fmin corresponds to a low-frequency
cutoff determined by the repetition rate of an experiment [84].
A comprehensive theoretical treatment of these different noise processes which dephase
charge qubits is presented in Refs. [53, 61, 64]. We will review both the charge noise and flux
noise contributions for the transmon qubit. In fact, in current transmon experiments, the
charge noise is sufficiently suppressed from operating in the transmon regime [63] that we
find flux noise to be the dominant culprit for dephasing.

Charge noise

In the Cooper pair box, with EJ /EC ≃ 1 the dephasing is primarily caused by slow fluctuations
of the offset charge ng , even while operating at the ‘sweet-spot’ [53, 60] which is first-order
insensitive to charge noise. The transmon qubit operates in a different regime, with EJ /EC ≫ 1,
resulting in the exponential suppression of the charge fluctuations. For 1/ f charge noise, the
typical amplitude is A = 10−4 − 10−5 e [85]. From Ref. [61] the dephasing time dependence on
72 superconducting qubits & cqed

small fluctuations of the charge dispersion is given by

−1
ħ ∂E01 ħ
Tϕ ∼ ∣ ∣ ≃ (3.58)
A ∂ng Aπ∣є1 ∣

where є1 is given by (3.14). Therefore, Tϕ exponentially increases with EJ /EC and becomes
very insensitive to charge fluctuations in the transmon regime. For typical sample parameters
of EJ = 25 GHz, EC = 350 MHz, gives a dispersion є1 ∼ 3 kHz and using A ∼ 10−4 , we find
dephasing times due to charge noise to be Tϕ ∼ 170 µs, far above 2T1 .

Flux noise

Whereas charge noise is effectively removed by operating with EJ /EC sufficiently in the
transmon regime, flux noise can still be a significant dephasing mechanism. Specifically,
noise in the externally applied flux can result in fluctuations of the effective Josephson
coupling energy EJ . Recall the external flux-dependent functional form for EJ (Φ̃) given in
(3.8)b. Here, EJ is periodic in Φ̃, and there are maximal values at which it becomes first-order
insensitive. These applied flux bias locations are ‘sweet-spots’ in δ Φ̃.
In Ref. [61], the effect of flux noise is computed at and away from flux sweet-spots. Away
from the sweet-spots, the dephasing time due to flux noise is

ħ ∂E01 −1 ħ Φ0 πΦ πΦ −1/2
Tϕ ≃ ∣ ∣ = (2EC EJmax ∣sin tan ∣) , (3.59)
A ∂Φ A π Φ0 Φ0
where the relevant noise parameter A has been found historically to be 10−5 Φ0 [86]. Figure 3.11
shows the frequency, relevant frequency slope versus flux, and inferred Tϕ for both A = 10−5 Φ0
and A = 10−6 Φ0 for EJ = 25 GHz and EC = 350 MHz. Simply detuning to Φ0 /4, results in
a Tϕ ∼ 1.5 µs for the 10−5 Φ0 case, which is now on the order of Purcell limited relaxation
times. However, at the flux sweet spot, the dephasing time can be estimated with the second
derivative of the transition frequency as

π 2 A2 ∂2 E01 −1 ħΦ20
Tϕ ≃ ∣ ∣ = √ (3.60)
ħ ∂Φ2 Φ=0 A2 π 2 2EJ EC

and with the same parameters as above will give Tϕ ∼ 40 ms.

Experimentally, we have found across multiple transmon qubit samples that detuning
away from the flux-sweet spots results in dephasing times that are in agreement with the
flux-noise treatment. We also find that the noise parameter A is in the range of 10−5 − 10−6 Φ0 .
 3.5. qubit decoherence 73

50 8
(a)
40
6
∂ω q /∂Φ (π ⋅ GHz/Φ )

ωq

ω q (π ⋅ GHz)
∂ω q /∂Φ
30
4
20

2
10

0 0
-4 8
10
(b)
6

ω q (π ⋅ GHz)
-5
10
Tϕ (s)

4
-6
10 ωq
2
Tϕ , A = − Φ
Tϕ , A = − Φ
-7 0
10
0.0 0.2 0.4 0.6 0.8 1.0
Φ (Φ )

Figure 3.11: Tϕ due to flux noise. (a) The qubit frequency ω q (solid black) and its derivative
with respect to flux ∂ω q /∂Φ (solid red) are shown versus flux Φ. (b) Assuming a flux noise
spectrum amplitude A = 10−6 Φ0 , the inferred flux-dependent Tϕ (solid green) is plotted versus
Φ.
74 superconducting qubits & cqed

Sensitivity to changes in the external flux (moving dewars, magnetic shoes) is reduced in
the experimental setup by using cryogenic magnetic shielding (detailed further in chapter 5).
Most experiments currently aim to operate the qubits at their flux sweet-spots in order to
reduce the effect of the flux noise. Specifically in the experiments in chapter 8 and chapter 9,
the qubits are parked at their flux sweet-spots, where the coherence times are ∼ 1 − 2 µs, for
performing state initialization, single-qubit operations and a joint readout.

3.6 Chapter summary

In this chapter we have reviewed the superconducting charge-based transmon qubit and
discussed its coupling to a microwave resonator in circuit QED. We have also introduced
some of the basic regimes of circuit QED and made an association with the well-known cavity
QED architecture. Thus far we have only considered circuit QED with a single transmon
qubit. To build a simple quantum processor, we look to scale this up, and use the microwave
resonator as a quantum bus. The next chapter continues the treatment of circuit QED, but
associates the quantum information processing protocols of chapter 2 with the coupled qubit
and microwave cavity system described in this chapter.
 CHAPTER 4

Circuit QED: Quantum Information Processing with a

Photon Bus

T
he circuit QED architecture discussed in chapter 3 can be used as a quantum bus coupling
scheme for multiple qubits. As previously described, transmon qubits can be coupled
to a microwave coplanar waveguide resonator. We can investigate the case of having two
transmons, both of which can interact with a single quantum bus by being placed at opposite
ends of the microwave resonator. If we drive the λ/2 resonance, there are anti-nodes in
the voltage at both ends of the resonator, which give the strong electric dipolar coupling as
described in section 3.4. Therefore, we can consider a two-qubit cavity bus device, and try to
understand how to perform some of the quantum information processing protocols from
chapter 2.
Specifically, we will need to demonstrate a universal set of gate operations (section 2.1) as
well as a full quantum-state readout (section 2.5.2). This chapter will start with a discussion
about the initialization of qubits in circuit QED and the effect of residual cavity excitations
on the starting state of the transmon qubits (section 4.1). Next, we will describe how we
can incorporate microwave driving to implement single-qubit operations in circuit QED
(section 4.2). Then, in section 4.3 we will discuss how the photon quantum bus provides
a number of interactions that will be useful for implementing multi-qubit entanglement
operations. These interactions can be turned on and off through fast tuning of the qubit

75
76 cqed: qip with a photon bus

transition frequencies. Although we will be performing quantum logic treating the transmons
as simple qubits, we will describe an interesting two-qubit interaction involving the higher
excited levels (section 4.3.3). Finally, in section 4.4, we will expand on the single-qubit
dispersive readout described previously (section 3.4.3) and present a joint two-qubit readout
mechanism, where the same bus used for multiple qubit coupling can also be used for their
multiplexed state detection.

4.1 Initialization

For the purposes of initializing a quantum register of m qubits in a circuit QED system, we
require two starting conditions:
( j)
1. Qubits dispersively detuned from cavity, ∣ωq − ωC ∣ ≫ , ∀ j ∈ m and qubits detuned
from one another by larger than the transmon anharmonicity to the second excited
( j)
state, ∣ωq − ωq ∣ ≫ α1 , ∀i, j ∈ m.
(i)

2. Mean cavity photon number exceedingly small, n̄ ∼ 0, and all intrinsic, non-cavity,
transition processes take the qubit to the ground state.

Condition 1 imposes the dispersive regime of circuit QED, with the qubit transitions separated
from one another in frequency space. Furthermore, this condition also avoids possible
excitations due to virtual photon exchange with the cavity (see section 4.3.2). In addition,
different qubit transition frequencies will allow independent driving of the qubits for single
qubit rotations (see section 4.2). The computational basis states will then be very simple,
{∣k1 , k2 , ..., k m ⟩}, where k i ∈ {0, 1}. For an m-qubit register, there are thus 2m basis states.
Condition 2 enforces that the qubit register will start from the joint ground state, ∣0, 0, ...0⟩.
The mean photon number in the cavity n̄ is directly related to the temperature of the bath to
which it is connected. Assuming a bath temperature T, the n̄ at the cavity frequency is given
by the Bose-Einstein distribution,
1
n̄ = . (4.1)
exp(ħωC /k B T) − 1

For typical experimental situations, T = 15 − 50 mK, corresponding to dissipative 50 Ω

attenuators and microwave circulators anchored to the base temperature of the dilution
fridge (more details of the experimental setup in chapter 5). In the case of our experiments,
n̄ ≲ 0.003 [89] (further details will be given in chapter 6).
 4.2. single-qubit gates in circuit qed 77

Although this is the average excitation number for a photon only at the single cavity
frequency ωC , we also need the second part of condition 2 to hold such that the dominant
excitation mechanism would be via the cavity, or that all other reservoirs to which it may
decay are at the same or lower temperature than the cavity. As discussed in section 3.5.1
and in detail in Ref. [82], the relaxation of the transmon qubits can be mostly attributable to
spontaneous emission through the cavity, γκ , via the multi-mode Purcell effect. However,
also recall that there is an intrinsic loss due to a constant Q ∼ 50, 000 − 70, 000 which sets
in at qubit transition frequencies such that γκ < ωq /Q. At this time the temperature of the
reservoir to which this unknown loss mechanism is connected to is neither characterized
nor known. Nonetheless, what we can say is that at locations where the qubit is multi-mode
Purcell limited, the qubit resets through the emission of a photon in equilibrium with a very
cold reservoir characterized by n̄ ≲ 0.003. Assuming that the qubit is only in equilibrium
with this photon bath, the residual excited state polarization, P1 , will be bounded bt 0.003,
giving an initial qubit polarization in the ground state, P0 of at least 99.7%.
A strong-driving experiment while in the strong coupling regime of circuit QED can
be used with precise master-equation simulations to determine the mean photon number
[89]. This experimental demonstration of the initialization also provides a detailed view of
Jaynes-Cummings physics and will be discussed in chapter 6. For the purposes of this chapter,
from hereforth we assume that our initial state will be in the mutual ground state of all the
qubits, ∣0, 0, ..., 0⟩.

4.2 Single-qubit gates in circuit QED

In this section we will develop the groundwork for single-qubit gates, with rotations around
the three Cartesian axes of the Bloch sphere (section 2.2). A simple microwave drive with
controllable phase can be used for x and y rotations whereas either an off-resonant drive
which induces an ac-Stark shift or fast flux tuning can be used to perform direct z rotations.

4.2.1 Introducing a drive

We can include a separate external drive term to the Jaynes-Cummings Hamiltonian of (3.40).
Consider multiple classical time-dependent coherent drives with a Hamiltonian

t
Hdrive = ∑(a + a † ) (ξ k e −iωd + ξ∗k e iωd t )
(k) (k)
(4.2)
k
78 cqed: qip with a photon bus

(k)
where ξ k is the strength, ωd is the frequency of the kth drive, and a and a† are the cavity
annihilation and creation operators. When the drive strengths are weak compared to the
other relevant energies (ωC , ) the rotating-wave approximation (RWA) can be applied to
give,

Hdrive = ∑ aξ∗k e iωd t

+ a † ξ k e −iωd t .
(k) (k)
(4.3)
k

For the moment, let us consider the case of just a single drive on a single qubit in circuit QED.
When combined with (3.40), we can remove the time-dependence of the full Hamiltonian
by making the following unitary transformation to enter the rotating frame of the drive:
⎡ ⎤
⎢ ⎥
U(t) = exp ⎢ iωd t(a a + ∑ ∣ j⟩ j ⟨ j∣)⎥⎥ .
⎢ †
(4.4)
⎢ ⎥
⎣ j
⎦
The full Hamiltonian now takes the form

H =U(H + Hdrive )U † − iU U̇ † (4.5a)

=ħ(ωC − ωd )a † a + ħ ∑ [(ω j − jωd ) ∣ j⟩ ⟨ j∣ +  j, j+1 (a† ∣ j⟩ ⟨ j + 1∣ + a ∣ j + 1⟩ ⟨ j∣)]
j

+ (aξ∗ + a † ξ)
=ħ∆r a † a + ħ ∑ ∆ j ∣ j⟩ ⟨ j∣ + ħ ∑  j, j+1 (a † ∣ j⟩ ⟨ j + 1∣ + a ∣ j + 1⟩ ⟨ j∣)
j j

+ (aξ(t)∗ + a † ξ(t)) , (4.5b)

where we have introduced the frequency differences ∆r = ωC − ωd and ∆ j = ω j − jωd and

allowed for the drive strength to be a slow function of time ξ(t).
The drive here still reflects transitions to the cavity. We can instead enter a frame such
that the drive Hamiltonian acts on transitions of the transmon by performing a Glauber
displacement transformation (details in Refs. [62, 79]), with displacement operator given by

D(α) = exp [α(t)a † − α ∗ (t)a] . (4.6)

By choosing α(t) as the solution to the differential equation

−i α̇(t) + ∆r α(t) + ξ(t) = 0, (4.7)

 4.2. single-qubit gates in circuit qed 79

the Hamiltonian takes the simple form

H = ∆r a † a + ∑ ∆ j ∣ j⟩ ⟨ j∣ + ∑  j, j+1 (a† ∣ j⟩ ⟨ j + 1∣ + a ∣ j + 1⟩ ⟨ j∣)

j j
1 (4.8)
+ ∑ (Ω∗ (t) ∣ j⟩ ⟨ j + 1∣ + Ω(t) ∣ j + 1⟩ ⟨ j∣) ,
2 j

where Ω(t) = 2α(t) gives the Rabi frequency. For a time-independent drive, this Rabi
frequency is given by Ω = 2ξ/∆r . Here in the limit where the detuning ∆r is large compared
to the cavity half-width κ/2, we can write the average photon number n̄ = (ξ/∆r )2 such that
√
the Rabi frequency recovers the Jaynes-Cummings form Ω ≈ 2 n̄.
Now going to the dispersive regime as described in section 3.4.2, the Hamiltonian is given
by

H/ħ = ∑ ω j ∣ j⟩ ⟨ j∣ + ωC a † a + ∑ χ j, j+1 ∣ j + 1⟩ ⟨ j + 1∣ − χ01 a † a ∣0⟩ ⟨0∣

j j

+ ∑ (χ j−1, j − χ j, j+1 ) a† a ∣ j⟩ ⟨ j∣ (4.9)

j=1
1
+ (Ω∗ (t)c + Ω(t)c † ) .
2
If we treat the transmon as just a qubit, we can write the driven dispersive JC using Pauli
matrix notation,
ħ
H= ∆q σz + ħ (∆r + χσz ) a † a + (Ω∗ (t)σ− + Ω(t)σ+ ) , (4.10)
2
where we see the first term as a re-normalized qubit, the second term reflects a qubit state
dependent cavity, and the third term is time-dependent raising and lowering of the qubit
excitation.

4.2.2 X-Y gates for a qubit

Staying in the qubit approximation of the transmon and assuming a drive of the form
⎧
⎪
⎪
⎪Ω x (t) cos(ωd t) + Ω y (t) sin(ωd t), 0 < t < t
Ω(t) = ⎨ (4.11)
⎪
⎪
⎪
⎩0, otherwise,
80 cqed: qip with a photon bus

which has two independent quadrature controls, Ω x (t), and Ω y (t), with t  being the total
time for a gate to take place, (4.9) can be re-written as

ħ ħ
H= ∆q σz + ħ (∆r + χσz ) a† a + (Ω x (t)σx + Ω y (t)σ y ) . (4.12)
2 2
Since we apply the drive far from the frequency band ωC ± χ where the cavity population
can be significant, the Lorentzian transmission damps the average photon number, giving
⟨a† a⟩ ∼ 0. Now by choosing the detuning between the drive and the qubit frequency to be 0,
∆q = 0, the above Hamiltonian generates rotations either around the x or y axes depending
on the choice of Ω x (t) and Ω y (t). For example, choosing a drive Ω x = Ω π and Ω y = 0, which
t
is on for a time t  , with ∫0  Ω π dt = π, will be a π-pulse, or bit-flip gate σx , that takes the qubit
population from the ground state to the excited state and vice versa. The π-pulse can similarly
be performed as a y-rotation just by switching, Ω y = Ω π and Ω x = 0. Moreover, π/2 pulses
√
around x and y can also be performed to make superpositions of the qubit, (∣0⟩ ± ∣1⟩)/ 2
√
and (∣0⟩ ± i ∣1⟩)/ 2, respectively. Combinations of x and y rotations can be used to perform
arbitrary rotations about any axis.

4.2.3 X-Y gates for a transmon multi-level atom

In practice, the transmon is not a simple two-level qubit, as discussed in section 3.1.3, but a
rich anharmonic multilevel system. It is a generalized atom which is subject to the problem
of leakage out of the qubit subspace. This is especially an issue when the bandwidth of
the control is comparable to the anharmonicity. For the case of the transmon, the typical
anharmonicity between the second excited state and the first excited state is

α1 → −EC (4.13)

for large EJ /EC . Typical design parameters result in charging energies ∼ 300 − 400 MHz,
which can be on par with the bandwidth of the shortest experimentally applied microwave
control pulses (∼ 1 − 2 ns).
In Ref. [90], a proposal to reduce higher-level leakage due to shorter control pulses is
presented in which optimized control pulses can permit high fidelity single-qubit gates. The
system is described with a truncation to three levels, from which an error due to the leakage to
the third level can be interpreted as a phase accumulation within the two-level qubit subspace.
For simplicity, we drop the cavity and consider a single qubit with three levels and a drive
 4.2. single-qubit gates in circuit qed 81

given by the Hamiltonian

H = ħ ∑ [ω j ∣ j⟩ ⟨ j∣ + ξ(t)λ j (σ j+ + σ j− )] , (4.14)
j=1,2

where σ j− = ∣ j − 1⟩ ⟨ j∣ and σ j+ = ∣ j⟩ ⟨ j − 1∣ are lowering and raising operators, ħω j denote

transition energies with the ground state energy set to zero, and
⎧
⎪
⎪
⎪1, j=1
λj = ⎨ (4.15)
⎪
⎪
⎪ j = 2,
⎩ λ,
is a weighting parameter on the relative strength between the 0-1 and 1-2 transitions. The
anharmonicity of this reduced system α1 is given by ω2 − 2ω1 .
Now we allow quadrature control again over the drive, resulting in an effective driven
Hamiltonian given by

Ω x (t) λΩ x (t)
H = ħ ∑ ∆ j ∣ j⟩ ⟨ j∣ + (∣0⟩ ⟨1∣ + ∣1⟩ ⟨0∣) + (∣1⟩ ⟨2∣ + ∣2⟩ ⟨1∣)
j=1,2 2 2
(4.16)
Ω y (t) λΩ y (t)
+ (∣0⟩ ⟨1∣ + i ∣1⟩ ⟨0∣) + (∣1⟩ ⟨2∣ + i ∣2⟩ ⟨1∣) .
2 2
Ideally λ = 0, and (4.16) recovers the single-qubit driven Hamiltonian as discussed previously
(4.9). However, more generally when λ is nonzero, leakage out of the qubit subspace will be
dictated by the bandwidth of Ω x (t) and Ω y (t) in comparison to the anharmonicity α1 . To
quantify the leakage to the third level, one can apply an adiabatic transformation V ,

V (t) = exp [−iΩ x (t)(∣1⟩ ⟨0∣ − i ∣0⟩ ⟨1∣ + λ(∣2⟩ ⟨1∣ − i ∣1⟩ ⟨2∣))/2α1 ] . (4.17)

The drive is turned on at t = 0 and off at t = t  , such that the effect of the applied pulses are
identical in both frames. By transforming the driven Hamiltonian of (4.16), we now have to
82 cqed: qip with a photon bus

first order in the small parameter Ω x /α1 ,

H/ħ = V HV † /ħ + i V̇ V †
Ωx λ(Ω x )2 (λ2 + 2)(Ω x )2
≈ (∣1⟩ ⟨0∣ + ∣0⟩ ⟨1∣) + (∣2⟩ ⟨0∣ + ∣2⟩ ⟨0∣) + (∆2 + ) ∣2⟩ ⟨2∣
2 8α1 4α1
(λ2 − 4)(Ω x )2
+ (∆1 − ) ∣1⟩ ⟨1∣
4α1
Ω y Ω̇ x
+( + ) [(∣1⟩ ⟨0∣ − i ∣0⟩ ⟨1∣) + λ(∣2⟩ ⟨1∣ − i ∣1⟩ ⟨2∣)] .
2 2α1
(4.18)

From this expression, we can see that a drive which performs a rotation around the x axis in
the simple two-level picture can actually result in both a phase error via a residual y-rotation
((∣1⟩ ⟨0∣ − i ∣0⟩ ⟨1∣) term), and a leakage to the second excited state (λ(∣2⟩ ⟨1∣ − i ∣1⟩ ⟨2∣) term).
However, this effect can be adiabatically eliminated by using the other quadrature by
setting

Ω̇ x
Ω y (t) = − (4.19)
α1
and furthermore, a phase shift error to the first excited state is removed by detuning the drive
such that
(λ2 − 4)(Ω x (t))2
∆1 = . (4.20)
4α1
Further corrections can be found by taking the transformation out to higher order. These
other terms are detailed in Ref. [90]. For the purposes of this thesis, we will be discussing
in chapter 6 the experimental implementation of this first-order correction by applying the
derivative of the drive on the quadrature during the pulse. This technique has been denoted
DRAG for Derivative Removal by Adiabatic Gate.
Experimentally, it has been common practice to shape the pulses with truncated Gaussian
(ΩG ) envelopes,

(t − t  /2)2
ΩG (t) = A exp [− ] [Θ(t) − Θ(t − t  )] (4.21)
2σ 2

where σ is the standard deviation for the Gaussian, A is determined by the amount of rotation
desired, and Θ(t) is the Heaviside function to indicate the truncation at t = 0 and t = t  . The
 4.2. single-qubit gates in circuit qed 83

1.2 σ = 1 ns, σ f = 159 MHz

σ =2, σ f = 80
1.0 σ =3, σ f = 53
σ =4, σ f = 40
0.8 σ =5, σ f = 32
Amplitude

σ =6, σ f = 27
0.6
σ =7, σ f = 23
σ =8, σ f = 20
0.4
σ =9, σ f = 18
0.2

0.0
5.6 5.7 5.8 5.9 6.0
Frequency (GHz)

Figure 4.1: Frequency bandwidth of Gaussian pulse shapes. For the Gaussian pulse shapes
given in (4.21), the fourier transform can be taken to determine the frequency extent. Given a
Gaussian with standard deviation σ given in time, the equivalent frequency Gaussian standard
deviation is given by σ f = (2πσ)−1 . Here in this figure we can see that for σ = 1 ns, there can
be a significant σ f = 159 MHz which can cause unwanted errors via higher-order transmon
excitations.

Gaussian pulse-shape has been chosen as opposed to simple square pulses due to its small
frequency response bandwidth, minimizing the excitations at the transition frequency of
the second excited state. Figure 4.1 shows the bandwidth of the Gaussians for different pulse
lengths. We can see that for longer pulses, the frequency bandwidth can be much smaller
than the anharmonicity of the third level, which in standard practice is ∼ 300 − 450 MHz.
Nonetheless, at the shortest pulse lengths, such as σ = 1 − 2 ns the third-level effects can
become significant, as we will detail in chapter 6.
We can characterize the quality of a gate using the single-qubit gate fidelity, which is
defined as [90]
1
F = ∑ †
Tr[Uideal ρ j Uideal χ(ρ j )], (4.22)
6 j=±x,±y,±z

where Uideal is the unitary transformation in the three-dimensional Hilbert space correspond-
ing to the idealized gate, ρ j are the six axial states of the qubit Bloch sphere, and χ(ρ J ) is the
actual experimental process. More details about determining the process matrix will be given
later in this thesis in chapter 6 in regards to gate characterization protocols.
84 cqed: qip with a photon bus

100

10-2
Gate error, 1-Fg

10-4

10-6

10-8

3 4 5 6 7 8
Gate time tg (ns)

Figure 4.2: Error per gate with and without DRAG. Standard Gaussian pulses with standard
deviation σ, and total gate time t  = 2σ, result in an error per gate (blue solid line) which
increases with decreasing gate time (neglecting relaxation processes) due to leakage to the
second excited state of the anharmonic qubit spectrum. DRAG pulses result in an error per
gate which decreases (red dashed line) down to a minimum value of ∼ 10−6 , well at the fault
tolerant threshold. √
Simulation is performed assuming a drive coupling strength to the second
excited state of λ = 2 and anharmonicity α1 = 2π(−400 MHz) and no decoherence properties.
(Figure used with permission from [90]. See Copyright Permissions.).

Based off of Ref. [90], we can simulate the effect of the pulse shaping. Specifically, by
assuming no relaxation processes (for the purposes of seeing the effect of the shaping), drive
√
coupling strength λ = 2, and third level anharmonicity given by α1 = 2π(−400) MHz,
F  is limited to 99% when the total gate time t  is 6 ns and using standard Gaussian pulse
shaping. Figure 4.2 shows in blue the error per gate, defined as 1 − F  for Gaussian pulses
with a standard deviation chosen to be 0.5t  . However, by using DRAG for the pulse shaping,
1 − F  can be reduced to the curve in red, achieving a minimum gate error of ∼ 10−6 . An
experimental implementation of derivative pulse-shaping for single-qubit gates based on
DRAG will be discussed in chapter 6, which will show a similar improvement in gate fidelity.
 4.2. single-qubit gates in circuit qed 85

4.2.4 Z (phase) gates

Although concatenations of arbitrary x and y rotations are sufficient for traversing the entire
single-qubit Bloch sphere, it can also be useful to directly perform rotations about the z axis.
This is especially the case when the ratio of coherence times (T1 , T2 ) to the gate time (t  ) is
not very large, such that having full control over the three axes allows for an optimized gate
sequence which takes up less time.

ac-Stark gate

One option for performing a direct rotation on the z axis is by employing the off-resonant
ac-Stark shift effect. A drive which is sufficiently detuned from the qubit to not induce
direct transitions via the σx term (Rabi frequency is small ωd − ωq ≫ 2Ω) will shift the qubit
transition frequency due to virtual photon transitions. Starting with the driven transmon
circuit QED Hamiltonian of (4.8), we can obtain an effective Hamiltonian which removes
the effect of direct transitions via the drive by using the unitary transformation
⎡ ⎤
⎢ ⎥
U = exp ⎢∑ β j (∣ j + 1⟩ ⟨ j∣ − ∣ j⟩ ⟨ j + 1∣)⎥⎥
⎢ (4.23)
⎢ j ⎥
⎣ ⎦
where β j = Ω(t)/2∆ j . This effective Hamiltonian to second order in β j is then given by

H = U HU † (4.24a)
= ∆r a † a + (∆0 + η0 ) ∣0⟩ ⟨0∣ + ∑(∆ j + η j, j−1 ) ∣ j⟩ ⟨ j∣
j=1

+ ∑ ̃ j, j+1 (a ∣ j + 1⟩ ⟨ j∣ + a ∣ j⟩ ⟨ j + 1∣) ,
†
(4.24b)
j

with
Ω2 (2∆0 + ∆1 )
η0 = (4.25a)
4∆20
Ω2 (2∆ j + ∆ j+1 ) Ω2 (4∆ j−1 + ∆ j )
ηj = − (4.25b)
4∆2j 4∆2j−1
Ω2 Ω2 Ω2  j+1
̃ j, j+1 =  j (1 − − + ). (4.25c)
4∆2j 8∆2j+1 4∆ j ∆ j+1  j
86 cqed: qip with a photon bus

Next, we can follow this with another dispersive transform to second order, and we are left
with the Hamiltonian

˜ r a † a + ∑ ∣ j⟩ ⟨ j∣ (∆ j + η j ) + ∑ χ̃ j, j+1 ∣ j + 1⟩ ⟨ j + 1∣ − χ̃01 a † a ∣0⟩ ⟨0∣

H≈∆
j j

+ ∑ ( χ̃ j−1, j − χ̃ j, j+1 ) a† a ∣ j⟩ ⟨ j∣ (4.26a)

j=1

where the χ̃ i j are now calculated using ̃i j . Therefore, for the transmon, operating with an
off-resonant drive, the 0-1 transition frequency can be ac-Stark shifted by an amount η1 − η0 .
When taking just a two-level truncation of the transmon, the effective Hamiltonian is then

1 1 Ω2
H ≈ ∆r a † a + (∆q + χ + ) σz (4.27)
2 2 ∆q

and the last term can be used to produce controlled rotations about the z axis. Although this
is a useful procedure for shifting the phase of a single-qubit, note that when multiple qubits
are coupled to the same bus, each qubit will suffer a frequency shift even when the other
qubits are driven. Furthermore, for the transmon qubits, coupling to the higher levels cannot
necessarily be ignored, and the Stark shift can become non-linear with respect to power of
the drive due to different Stark shifts of the higher levels [79].

Flux gate

Another method for direct z rotations is to use the non-linear dependence of the qubit
transition frequency on the applied flux to shift the qubit transition frequency by a controlled
amount. Recall that the transmon Hamiltonian is given by

H = ħ ∑ ω j ∣ j⟩ ⟨ j∣ (4.28)
j

where the ω j are given in . However, in the transmon limit, where EJ ≫ EC and for a two-level
truncation, the qubit Hamiltonian is simply
√
H q = ħ 8EJmax ∣cos(πΦ/Φ0 )∣ EC σz . (4.29)
 4.3. two-qubit gates in circuit qed 87

In the dispersive regime the full Hamiltonian is still flux-tunable, and a controlled amount of
z-phase θ z can be obtained by controlling Φ over a gate period t  such that,
t √
θz = ∫ dt 8EJmax ∣cos(πΦ(t)/Φ0 )∣. (4.30)
0

In chapter 8, these flux-based z rotations will be an important part of tuning large amounts of
dynamical phase which are accumulated during a separate flux excursion used for a two-qubit
c-Phase gate.

4.3 Two-qubit gates in circuit QED

We now switch our attention to two qubits in a circuit QED system and focus on how to realize
entangling gates (section 2.3) to complete the suite of gates necessary for universal quantum
computing (section 2.1). A full treatment of two qubit gates is given in Ref. [91]. In this
section we will only highlight one of those gates, the virtual swap interaction (section 4.3.2).
Besides the virtual swap, there can also be an indirect swap between qubits by tuning into
direct interaction with the resonator, experimentally implemented in phase qubits [92, 93].
In circuit QED, it is also possible to use sideband transitions to perform two-qubit gates, in a
scheme similar to the coupling of trapped-ion qubits, and experimentally investigated for
generating Bell states [94], as well as multiple fixed off-resonant drives in a scheme called
FLICFORQ [95]. However, even with the numerous two-qubit entangling gates for circuit
QED described in Ref. [91], the other gate which we describe in section 4.3.3 is a completely
different approach that relies on the multi-level structure of transmon qubits. Later in this
thesis in chapter 8, we will demonstrate the experimental implementation of this two-qubit
gate and generate highly-entangled states.
88 cqed: qip with a photon bus

4.3.1 Two-qubits in the dispersive regime

With two transmons, the standard Jaynes-Cummings Hamiltonian is modified to include the
strong coupling of the photon field with both transmons,

H JC = ωC a† a + ∑ ω j ∣ j⟩1 ⟨ j∣1 + ∑ ω i ∣i⟩2 ⟨i∣2

(1) (2)

j i

+ ∑  j, j+1 (a † ∣ j⟩1 ⟨ j + 1∣1 + a ∣ j + 1⟩1 ⟨ j∣1 )

(1)
(4.31)
j

+ ∑ i,i+1 (a† ∣i⟩2 ⟨i + 1∣2 + a ∣i + 1⟩2 ⟨i∣2 ) .

(2)

To get to the dispersive regime, the resonant Jaynes-Cummings interaction is eliminated

using the transformation
⎡ ⎤
⎢ ⎥
U = exp ⎢∑ λ j (a ∣ j⟩1 ⟨ j + 1∣1 − h.c.) + ∑ λ i (a ∣i⟩2 ⟨i + 1∣2 − h.c.)⎥⎥
⎢ † †
(1) (2)
(4.32)
⎢ j ⎥
⎣ i
⎦

where λ j =  j, j+1 /(ω j+1, j − ωC ) =  j, j+1 /∆ j ≪ 1. To second-order in λ(k) , the two transmon
(k) (k) (k) (k) (k)

dispersive Hamiltonian is then given by

H/ħ =ωC a † a + ∑ (ω j ∣ j⟩1 ⟨ j∣1 + χ j, j+1 ∣ j + 1⟩1 ⟨ j + 1∣1 )

(1) (1)

+ ∑ (ω i ∣i⟩1 ⟨i∣1 + χ i,i+1 ∣i + 1⟩2 ⟨i + 1∣2 )

(2) (2)

− a† a (χ01 ∣0⟩1 ⟨0∣1 + χ01 ∣0⟩2 ⟨0∣2 )

(1) (2)

⎡ ⎤
⎢ ⎥
+ a a ⎢∑ (χ j−1, j − χ j, j+1 ) ∣ j⟩1 ⟨ j∣1 + ∑ (χ i−1,i − χ i,i+1 ) ∣i⟩2 ⟨i∣2 ⎥⎥
† ⎢ (1) (1) (2) (2)
⎢ j=1 ⎥
⎣ i=1
⎦
 j, j+1 i,i+1+ (∆ j + ∆ i )
(1) (2) (1) (2)

+∑ [∣ j⟩ ⟨ j + 1∣1 ⊗ ∣i + 1⟩ ⟨i∣2 + ∣ j + 1⟩ ⟨ j∣1 ⊗ ∣i⟩ ⟨i + 1∣2 ] .

2(∆ j ∆ i )
(1) (2)
ji

(4.33a)

If we consider both transmons as only qubits, the effective dispersive Hamiltonian is then
simplified to
1 1
H/ħ = (ωC + χ1 σz + χ2 σz ) a † a + ω1 σz + ω2 σz
(1) 2) (1) (2)
2 2
1 2 (∆1 + ∆2 ) (1) (2)
+ (σ+ σ− + σ− σ+ ) ,
(1) (2)
2∆1 ∆2
 4.3. two-qubit gates in circuit qed 89

where the first term is the cavity dispersively shifted by both qubits, the second and third
terms are the bare qubit Hamiltonians, and the final term is a two-qubit swap which occurs
via virtual interaction with the cavity.

4.3.2 Virtual qubit-qubit interaction

Working within the two-qubit subspace with four independent levels, ∣0, 0⟩, ∣0, 1⟩, ∣1, 0⟩, and
∣1, 1⟩, we can see that the virtual swap interaction of the dispersive Hamiltonian is of the form
√
described in section 2.3.3 such that we can realize an entangling iSWAP gate. We introduce
the parameter J to represent the strength of the swap interaction,

1 2 (∆1 + ∆2 )
J= . (4.34)
2∆1 ∆2
Now by operating in the dispersive regime, the cavity population can be small so ⟨a†a⟩ ∼ 0.
We now have a two-qubit unitary with the following time evolution,
−it
U2q (t) = exp [ (ω1 σz ⊗ 1(2) + ω2 1(1) ⊗ σz )]
(1) 2)
2
⎛1 0 0 0⎞
⎜ ⎟
⎜0 cos(Jt) i sin(Jt) 0⎟ (4.35)
⎜
×⎜ ⎟.
⎟
⎜0 i sin(Jt) cos(Jt) 0⎟
⎜ ⎟
⎝0 0 0 1⎠

The first piece is simply made up of single-qubit phases which are removable via the appro-
priate single-qubit rotations while the second piece in the large parantheses corresponds to
√
the iSWAP logical operation at t = π/(4J). The level diagram in figure 4.3 visually depicts
the virtual photon exchange with the cavity, from which we can see the cavity only acting as
a spectator to the interaction, never being actually populated with a photon.
Since the value of J governs the time for performing the entangling gate, the detuning
between the two qubits plays a critical role for the swap interaction. As the interaction term
σ+ σ− + σ− σ+ is energy swapping, we can see that when the qubits are not near resonance
(1) (2) (1) (2)

but far detuned, the swap will be suppressed. The maximal interaction occurs with the qubits
tuned into resonance with one another, such that ∆1 = ∆2 = ∆ and J = 1 2 /∆. The ability for
this interaction to be strong and weak depending on the qubit detuning provides a recipe
for operating a full set of universal qubit gates. One can detune the qubits for performing
single-qubit logical operations where the interaction is effectively off. Then, to perform the
90 cqed: qip with a photon bus

∣, ⟩ , n = 

ω + ω
∣, ⟩ , n =  ∣, ⟩ , n = 

∆
() () ∣, ⟩ , n = 
ω ω

ωC
∣, ⟩ , n = 

Figure 4.3: Scheme of the virtual photon swap interaction. When the qubits are detuned from
the cavity (∣∆(1),(2) ∣ =≫ (1),(2) ) the qubits both dispersively shift the cavity. The excited state
in the left qubit ∣1, 0⟩ ⊗ ∣n = 0⟩ interacts with the excited state in the right qubit ∣0, 1⟩ ⊗ ∣n = 0⟩
via the exchange of a virtual photon ∣0, 0⟩ ⊗ ∣n = 1⟩ in the cavity.

two-qubit logic gates, the qubits are tuned into resonance with one another for the appropriate
√
amount of time to realize the iSWAP.
√
To use the iSWAP gate as an two-qubit entangling gate reduces to being able to change
the detuning between the qubits on fast time-scales. The circuit QED architecture with
transmon qubits fortunately provides this tunability through either an off-resonant ac-Stark
shift or a fast dc-flux tuning.

Two-qubit ac-Stark swap gate

The off-resonant ac-Stark shift, which was introduced as a generator of rotations about the
z-axis in section 4.2.4, can also be used to tune the qubits in and out of resonance with one
another to effectively turn the swap interaction on and off. The treatment is similar to the
case of including an off-resonant drive, except now there are two transmons which each can
be Stark shifted differently due to their different interaction strengths and detunings.
The effect of applying a drive will is to shift the qubit transition frequencies for both of
the transmon qubits to

Ω2
ω̃ q = ω q + + χ̃(k) ,
(k) (k)
(4.36)
2∆ q

where χ̃ = χ̃01 − χ̃12 /2. The Stark swap gate can be performed by starting with the qubits
effectively uncoupled from one another, such that ∣ω q − ω(2) ∣ ≫ J, and then to turn on
(1)

the appropriate amplitude drive at a frequency ωd such that ω̃ q = ω̃ q , turning on the J

(1) (2)
 4.3. two-qubit gates in circuit qed 91

interaction. Although in the simplest two-level picture, the value of the Stark shift has a
simple relationship with the detuning of the drive from the qubit frequency, for transmon
qubits, higher-level couplings contribute to the Stark shift, especially at large drive powers
resulting in non-linear frequency shifts.
The operation of the Stark swap gate also relies on a low enough drive power that direct
transitions of the cavity do not occur, which would otherwise result in heating of the cavity
and an enhancement of decoherence. Another possible error is for the Stark shift to cause
direct transitions of the qubit due to insufficient detuning. The effective Rabi frequency of a
√
detuned drive is given by Ω′R = Ω2R + ∆2 , where Ω R is the Rabi frequency corresponding to
the applied drive power if it were on resonance with the qubit, ωd = ω q .
Furthermore, although the Stark effect can be an effective method for turning on a two-
qubit interaction, if the system expanded to more than two, there could be even higher-order
Stark shifts which can make the tunability unwieldy. An experimental implementation of
this Stark gate with two-qubits is described in chapter 7.

Fast flux swap gate

Another option for turning on the virtual flip-flop interaction is to directly tune the qubit
frequencies into resonance with one another using independent flux control on each of the
qubits. As described in section 4.2.4 in regards to the single-qubit phase gate via flux-tuning,
similarly the two qubits transition frequencies,
√
ω01 (t) = 8(EJ ∣cos(πΦ(i) (t)/Φ0 )∣ EC − EC ,
(i) max,(i) (i) (i)
(4.37)

can be tuned with a flux pulse such that at t = 0 with ω01 (0) ≠ ω01 (0) and at some later
(1) (2)

time t ′ they are tuned to be equal, ω01 (t ′ ) ≠ ω01 (t ′ ). The flux pulse rise-time needs to be
(1) (2)

faster than the swap rate, but still adiabatic with respect to the qubit transition frequencies.
Of course the ability for this gate to be used relies also on pre-determined device parameters,
such as the charging energies and maximum Josephson energies of both qubits. Nonetheless,
with fast independently tunable flux, this is a candidate for realizing the two-qubit entangling
swap gate, and can be extended to systems with more qubits. In chapter 8 we will show an
implementation of the swap using fast flux tuning.
92 cqed: qip with a photon bus

4.3.3 σz ⊗ σz higher level transmon interaction

The swap interaction discussed in the previous sections arises from the second order ex-
pansion of the Jaynes-Cummings Hamiltonian in the dispersive regime (4.3.1). By taking
the dispersive approximation to higher orders in λ j =  j, j+1 /(ω j, j+1 − ωC ), there are other
two-qubit interactions which emerge.
For example, we can consider the case of going to fourth order for a two-level approxima-
tion of both qubits. This can be done by using fourth-order perturbation theory. Although
all of the two-qubit interactions arising from this further expansion are much weaker than
the swap interaction, one which is of particular interest is a σz ⊗ σz term, an interaction
(1) (2)

which can be used to generate a c-Phase gate (section 2.3.2). The coupling strength of this
two-qubit interaction is given by ζ, involving a two-excitation process such that

12 22 (∆1 + ∆2 )

ζ= . (4.38)
∆21 ∆22

We can easily see that this coupling is smaller than J, by a factor of 1 2 /∆1 ∆2 . The relative
weakness of this interaction to the swap-interaction thus makes such a σz ⊗ σz interaction
(1) (2)

not very useful for performing a two-qubit gate. However, this situation changes significantly
when considering multiple levels in the transmons.
For a transmon qubit, the presence of higher levels can actually boost up the strength
of this interaction. Since the ζ is a result of a two-excitation process, specifically the second
excited state of the transmon qubit can also interact. Consider a set of two transmons, which
have a negative anharmonicity, arranged such that their single excitation transition energies
do not coincide, or ω10 ≠ ω01 . We will use the notation here that ω i j corresponds to the
transition energy for the two transmon state with the first transmon in state ∣i⟩ and the second
transmon in state ∣ j⟩. Now, suppose ω01 > ω10 . Then, by varying the applied flux on the qubit
with the higher single-excitation transition energy will generate an interaction in the two
excitation manifold, as shown in figure 4.4.
This interaction can be calculated using fourth order perturbation theory for a pair of
qutrits coupled to a cavity, and now takes a very different form from the simple two-level
 4.3. two-qubit gates in circuit qed 93

∣, ⟩ (a)
Frequency (GHz)

∣, ⟩

∣, ⟩

Φ

∣, ⟩ + ∣, ⟩
(b)
∣, ⟩
Frequency (GHz)

∣, ⟩

ζ
∣, ⟩

Φ

Figure 4.4: Two excitation manifold. (a) By varying Φ2 , the external flux on the transmon with
the higher EJ , the transition energy level corresponding to ∣0, 2⟩ can be tuned into an avoided
crossing with ∣1, 1⟩, where both transmons are in the first excited states. (b) Zoom-in of the
avoided crossing region shows the deviation of the transition energy of ∣1, 1⟩ (solid purple) from
the sum of the transition energies of ∣0, 1⟩ and ∣1, 0⟩ (solid black). This interaction strength of ζ
is the generator of a two-qubit conditional phase interaction. These simulations are performed
via numerical diagonalization of a Jaynes-Cummings model with two transmons, assuming
EJ1 = 28 GHz, EJ2 = 42 GHz, EC1 = 320 MHz, EC2 = 300 MHz, ωC = 7, GHz, and κ = 1 MHz.
94 cqed: qip with a photon bus

∣, ⟩

∣, ⟩

ζ
Frequency (GHz)

∣, ⟩

∣, ⟩

Φ
Figure 4.5: Level scheme for c-Phase. Theoretical level scheme shows that by varying the
magnetic flux Φ2 , it is possible to vary the single-qubit phase of the right qubit, θ z01 . Similarly,
varying the magnetic flux Φ1 can be used to change θ z10 . Due to the presence of ∣0, 2⟩ interacting
with ∣1, 1⟩, changing Φ2 can thus vary the phase on ∣1, 1⟩ in a way differently than from ∣0, 1⟩. As
a result, θ z11 ≠ θ z01 + θ z 10, and unitary operations such as the c-Phase gate are possible. These
simulations are again performed via numerical diagonalization of a Jaynes-Cummings model
with two transmons, assuming EJ1 = 28 GHz, EJ2 = 42 GHz, EC1 = 320 MHz, EC2 = 300 MHz,
ωC = 7, GHz, and κ = 1 MHz.
 4.3. two-qubit gates in circuit qed 95

case,

1 1
ζ = −212 22 ( +
(ω01 − ω12 )(ω01 − ωC )2 (ω01 − ω12 )(ω01 − ωC )2
(2) (1) (1) (1) (R) (2)
(4.39)
1 1
+ + ).
(ω01 − ωC )(ω01 − ωC )2 (ω01 − ωC )(ω01 − ωC )2
(1) (2) (2) (1)

This expression, we can see, diverges when the transition between the 0 and 1 levels of one
transmon aligns with the 1 and 2 levels of the second transmon. Therefore, a resonance in
the two excitation manifold can result in a much stronger σz ⊗ σz interaction.
(1) (2)

This interaction results in an avoided crossing between the ∣1, 1⟩ and ∣0, 2⟩ two-transmon
states. In terms of the qubit computational basis states, {∣0, 0⟩ , ∣0, 1⟩ , ∣1, 0⟩ , ∣1, 1⟩} the fre-
quency shift is completely on the ∣1, 1⟩ state, such that the interaction strength can be expressed
as

ζ = ω11 − ω01 − ω10 . (4.40)

The action of the shift can be directly used to alter the phase (figure 4.5) of the computational
basis state ∣1, 1⟩. As discussed in section 4.2.4, a fast flux pulse can be used to change the
frequency of the transmon transition frequency, resulting in a phase θ z given by (4.30). In the
case of two transmons, flux pulses can be used to modulate the phase of all the computational
basis states, giving access to a unitary transformation of the form,

⎛1 0 0 0 ⎞
⎜ ⎟
⎜0 e iθ z
01
0 ⎟
⎜
U =⎜
0 ⎟,
⎟ (4.41)
⎜0 0 e iθ z 0 ⎟
10

⎜ ⎟
⎝0 0 0 e iθ z ⎠
11

where the dynamical phase shifts are given by

ij
θ z = ∫ δω i j (t)dt. (4.42)

Now suppose a square shaped flux pulse is turned for time t f on one of the transmons such
that the interaction between the ∣1, 1⟩ and ∣0, 2⟩ gives
tf
∫0 ζ(t)dt = (2n + 1)π, (4.43)
96 cqed: qip with a photon bus

where n is an integer. As a result of (4.40), the phase shift of the ∣1, 1⟩ state can now be
expressed as

θ z11 = θ z01 + θ z10 − (2n + 1)π, (4.44)

resulting in a total two-qubit phase unitary given by

⎛1 0 0 0 ⎞
⎜ ⎟
⎜0 e iθ z
01
⎟
⎜
U =⎜
0 0 ⎟.
⎟ (4.45)
⎜0 0 e iθ z ⎟
10

⎜ 0 ⎟
⎝0 0 0 −e i(θ z +θ z ) ⎠
01 10

Now by simply tuning individual qubit phases with small frequency excursions using each
qubit’s independent flux bias, the other phases can be tuned such that θ z01 = θ z10 = 0, and we
are left with a two-qubit c-Phase (section 2.3.2) entangling gate,

⎛1 0 0 0⎞
⎜ ⎟
⎜0 0 0⎟
U =⎜ ⎟.
1
⎜ ⎟ (4.46)
⎜0 1 0⎟
⎜ 0 ⎟
⎝0 0 0 −1⎠

However, it is important to stress again that this σz ⊗ σz interaction for the c-Phase
(1) (2)

gate is only possible as a result of the avoided crossing between computational with non-
computational states. It is a technique that cannot work in simple two-level systems but is
accessible in any qubit implementation with finite anharmonicity, such as transmons or phase
qubits [96]. This permits the ζ to be much larger than the case of just a simple two-level
system, as previously noted in (4.38). Note also that it is the negative anharmonicity of the
transmons which permits this interaction to occur before the onset of the swap interaction J
by simply tuning the flux bias of one of the transmons.
One of the more subtle features of using this higher-level transmon interaction for gener-
ating an entanglement gate is the adiabatic flux tuning. Whereas the transverse swap coupling
discussed previously requires a fast-tuning of the qubit transition frequencies directly into
the avoided crossing, the σz ⊗ σz interaction requires only a slow tuning, acquiring phase
(1) (2)

throughout the adiabatic frequency shift. For experimental purposes, an adiabatic pulse can
be simpler to implement than a very fast and sudden pulse. Typical control pulse rise times
are on the order of 1 to 2 ns. Swap interactions which are J/2π ∼ 100 MHz have a period of
 4.4. muliplexed joint qubit readout 97

1 ∼ 2 ns, such that pulses do not necessarily turn on fast with respect to J. However, the pulses
need not be fast for the ζ interaction.
As discussed in chapter 2, the c-Phase gate combined with single qubit rotations form
a universal set (section 2.1) of gates for quantum computing protocols. Also recall that the
c-Phase gate is easily converted into to the more traditionally studied cNOT gate section 2.3.2.
Experimental implementation of the entanglement gate for producing Bell states and quantum
algorithms will be presented later in this thesis in chapters 8 and 9.

4.4 Muliplexed joint qubit readout

In section 3.4.2 we introduced the strong dispersive regime of circuit QED as a feature for
qubit-state determination via transmission through the cavity. By applying to the cavity a
microwave field close to its resonance frequency, the transmitted amplitude is a non-linear
function of the cavity pull resulting in a projective QND readout. The state dependent shift is
governed by χ, as a result of the dispersive ac Stark effect. If we extend the strong dispersive
regime to multiple qubits, the cavity Hamiltonian is now given by

H cav = a† a (ωC + ∑ χ(k) σz )

(k)
(4.47)
k

with k indexing the qubit number in an n-qubit register. If all of dispersive frequency
pulls are large with respect to κ and different from one another, then each of the 2n qubit
computational basis states {∣000...0⟩ , ∣100...0⟩ , ∣010...0⟩ , ..., ∣111...1⟩} will have a different
transmission frequency.

4.4.1 Deriving the measurement operator

For simplicity of notation, we can deal with just two qubits, although note that the treatment
can be extended for an arbitrary number of qubits. Two qubits in the strong dispersive regime
will have cavity peaks at the four peaks ωC + χ1 + χ2 , ωC + χ1 − χ2 , ωC − χ1 + χ2 , and ωC − χ1 − χ2 ,
corresponding to the states ∣0, 0⟩, ∣0, 1⟩, ∣1, 0⟩, and ∣1, 1⟩ respectively (figure 4.6). Now applying
a drive would query for a joint property of the entire two-qubit state. For example, by driving
at the frequency corresponding to ωC + χ1 + χ2 with the qubits in their ground state ∣0, 0⟩
interrogates the system, by detecting a transmitted signal only if both qubits are in the ground
state.
98 cqed: qip with a photon bus

1.0
∣, ⟩ ∣, ⟩ ∣, ⟩ ∣, ⟩
Transmission

χ() − χ()

χ() + χ()

ωC
0.0
6.94 6.96 6.98 7.00 7.02 7.04 7.06
Frequency (GHz)
Figure 4.6: Transmission in strong dispersive regime for two qubits. The four dispersive
shifted cavity transmission peaks for a two-qubit and cavity system with a bare transmission
frequency of 7 GHz, and qubits with dispersive shifts χ(1) /2π = 40 MHz and χ(2) /2π = 15 MHz.

A query of the cavity at the peak for ∣0, 0⟩ then gives rise to a measurement operator that
is a projector on the two-qubit ground state,

M = ∣0, 0⟩ ⟨0, 0∣ . (4.48)

This can expressed in terms of the Pauli matrices for the two qubits as

M = σz + σz + σz ⊗ σz .
(1) (2) (1) (2)
(4.49)

In practice however, dynamics of the qubits and the cavity during measurement give a more
generalized averaged measured observable given by

⟨M⟩ = β1 ⟨σz ⟩ + β2 ⟨σz ⟩ + β12 ⟨σz ⊗ σz ⟩,

(1) (2) (1) (2)
(4.50)

where β1 , β2 , and β12 are all ≤ 1 and reflect the sensitivity of the measurement to each of three
Pauli operators. We adopt a simpler notation when discussing the measurement of two-qubit
operators by identifying σz ⊗ 1(2) → ZI, 1(1) ⊗ σz → IZ, σz ⊗ σz → ZZ, and similarly
(1) (2) (1) (2)

for all other Pauli operators.

To understand why our measurement operator is given by (4.50), we start with the cavity
observables which correspond to our measurement, I = a† + a and Q = i(a † − a). If we
 4.4. muliplexed joint qubit readout 99

include a single drive, our Hamiltonian is

1 ( j) ( j)
H/ħ = χa † a + ∑ (ω q + χ j ) σz + є(t) (a † + a) (4.51)
2 j=1,2

( j)
where є(t) is the measurement amplitude and χ = ωC − ωd + ∑ j χ j σz . Our measurement
records ensemble averages of the quadrature voltage amplitudes, given by

⟨I⟩(t) = Tr [ρ(t) (a † + a)] (4.52a)

⟨Q⟩(t) = Tr [iρ(t) (a † − a)] . (4.52b)

From the Heisenberg dynamics and input-output theory [34, 97], the time-dependence of
the cavity annihilation operator can be found to be
√
ȧ = −i χa − iє − κa/2 − κb in (4.53)

where b in describes the photon bath field annihilation operator connected to the input of the
cavity. The solution to this expression is given by
κ

[−( κ2 +i χ)t]
(1 − e [−( 2 +i χ)t] )
a(t) = a(0)e −є . (4.54)
χ − iκ/2

We can now evaluate the expressions for the measured quadrature voltage operators by
assuming an initially separable state between the qubits and resonator, e.g.ρ(0) = ∣0⟩ ⟨0∣ ⊗
ρ q (0) where the resonator is in the ground state ∣0⟩ and the qubit density matrix for an
arbitrary state can be written as

ρ q (0) = ∑ p i ji ′ j′ ∣i j⟩ ⟨i ′ j′ ∣ . (4.55)
i, j,i ′ , j′ =0,1

Putting these expressions together, we can write the average values of the field quadratures as

є
⟨I⟩(t) = ⟨ [−2χ + e −κt/2 (2χ cos[χt] + κ sin[χt])]⟩ (4.56a)
χ2 + κ 2 /4
є
⟨Q⟩(t) = ⟨ [−κ − e −κt/2 (2χ sin[χt] − κ cos[χt])]⟩ , (4.56b)
χ2 + κ 2 /4
100 cqed: qip with a photon bus

which in the steady state are given by

−2χє
⟨I⟩(t) = ⟨ ⟩ (4.57a)
+ κ 2 /4
χ2
−χє
⟨Q⟩(t) = ⟨ ⟩. (4.57b)
χ2 + κ 2 /4

This result recovers the standard Lorentzian behavior of the power transmitted, however, as a
non-linear function of the two-qubit state-dependent cavity pull,

є2 є2
⟨M⟩ = I 2 + Q 2 = = . (4.58)
χ2 + κ 2 /4 (ω − ω + χ σ (1) + χ σ (2) )2 + κ 2 /4
C d 1 z 2 z

The ensembled measurement operator can then be decomposed in the two-qubit Pauli basis
set as

⟨M⟩ = β0 ⟨II⟩ + β1 ⟨ZI⟩ + β2 ⟨IZ⟩ + β12 ⟨ZZ⟩, (4.59)

where the coefficients can be found from partial traces, e.g. β ZI = Tr[M(t)ZI]. The coeffi-
cients β will depend on the drive frequency. In figure 4.7, we plot the coefficients for the two
quadratures I and Q as a function of ∆C = ωC − ωd for a cavity with linewidth κ, χ(1) = 10κ
and χ(2) = 2κ. We can see that by applying a drive at the bare resonator frequency, all of the β
coefficients are small, giving very little information about the ZZ correlator. However, driving
at the frequency corresponding to the ground state peak ∣0, 0⟩ will actually give maximal
values for all the β, particularly in the Q channel. As a result, direct access to two-qubit
correlations are attainable depending on the choice of drive. One point to note, however, is
that the treatment presented here does not take into account the decoherence effects of the
qubits during the drive for measurement. As the qubit relaxation rates are on the order of the
cavity decay, any pulsed interrogation drive will experience a bias of the qubit state towards
the ground state ∣0, 0⟩. Such an effect results in simply a re-normalization of the β values,
with the largest signal-to-noise in β obtained with a drive which is at the ∣0, 0⟩ measurement
peak. In chapter 8, we will see this in an experiment which characterizes this joint readout
and the dispersive shifts associated with a two qubit circuit QED device.
 4.4. muliplexed joint qubit readout 101

0.4

0.0
βI

-0.4

1.0 ∣, ⟩

0.5
βQ

0.0

-0.5

-1.0
-10 -5 0 5 10
∆C (κ)

Figure 4.7: Measurement model coefficients versus drive frequency. The measurement model
coefficients for ZI (blue), IZ (purple), and ZZ (gold) as a function of detuning ∆C from the bare
cavity frequency. Here the measurement model is broken down into the two cavity quadratures,
I and Q. The cavity is assumed to have a linewidth κ, and the simulation is performed assuming
χ(1) = 10κ and χ(2) = 2κ.

4.4.2 State tomography in circuit QED

As discussed in chapter 2, quantum state tomography (section 2.5.2) is the technique by which
the density matrix ρ of an arbitrary quantum state can be estimated. Finding this density
matrix is critical in any quantum computing architecture for determining how good the
quantum states are, and for the quantification of entanglement for entangled states.
The full determination of the state is traditionally performed by counting coincidences
between multiple individual qubit readouts [11, 15, 67, 98]. Often these individual readouts
will give ‘clicks’ corresponding to a single Pauli operator, e.g. σz . However, measurements
of the two-qubit state need to be performed in the combined bases of the two qubits, such
that there are nine combinations of the three Cartesian directions (x, y, z) for each qubit.
102 cqed: qip with a photon bus

Single-qubit rotations such as R y (π/2) and R x (π/2) can be performed prior to readout of
each qubit in order to rotate the measurement basis from σz to σx or σ y respectively. Hence,
two-qubit correlations are actually obtained through simultaneously recorded events on
each individual qubit readout combined with a classical product. However, such a readout
scheme is subject to scrutiny when readout fidelity is low, making the probabiliy of incorrect
counts much higher. Another issue is the presence of crosstalk between readouts. Although
this is not a serious problem in quantum systems such as photons and trapped-ions, in
which the individual detectors are simply photodiodes that can be spatially separated and
uncorrelated, for superconducting architectures, crosstalk can be quite impactful due to stray
electromagnetic coupling on a lithographically defined chip [67, 99].
Similar to the individual readouts, the joint readout discussed in this chapter can also be
used for full quantum state tomography. Since the ensembled measurement from (4.59) is a
function of not only individual qubit polarizations but also two-qubit correlations, by com-
bining a set of measurements involving single-qubit rotations before the joint measurement,
the state ρ can be reconstructed. Of course, this relies on a believable characterization of the
joint readout and the underlying measurement operator M. To satisfy this, the coefficients β
can be determined and calibrated by simple Rabi driving of the qubits between the different
computational basis states. This experiment will be described in detail in chapter 8.
Nonetheless, with the coefficients β determined, ensembles of all the two-qubit Pauli
operators can be found by applying the appropriate pre-rotations (chosen from R y (±π/2),
R x (±π/2), R x,y (π), 1) on both qubits. We can drop β0 as an overall offset constant. As an
example, we can get the ensemble average ⟨IZ⟩ by combining the measurements where we
apply 1(1) ⊗ 1(2) with the measurement where we apply R x (π) ⊗ 1(2) :
(1)

⟨M⟩1 = β1 ⟨ZI⟩ + β2 ⟨IZ⟩ + β12 ⟨ZZ⟩ (4.60a)

⟨M⟩2 = −β1 ⟨ZI⟩ + β2 ⟨IZ⟩ − β12 ⟨ZZ⟩. (4.60b)

And similarly, a set of fifteen two-qubit Pauli expectation values can be obtained to reconstruct
the full density matrix of the two-qubit state. Details of the full estimation and experimental
state tomography will be given in chapter 8.
The joint readout method for state estimation is fundamentally different from individual
qubit readouts, as the issues of poor single-shot readout fidelity and measurement crosstalk are
essentially circumvented. The single-shot readout fidelity for joint readout enters statistically
when it is poor and dominated by amplifier noise. Then, the ensemble averaged measurement
 4.4. muliplexed joint qubit readout 103

operator is subject to classical Gaussian fluctuations in the voltage, δv, and given by

⟨M⟩ = β1 ⟨ZI⟩ + β2 ⟨IZ⟩ + β12 ⟨ZZ⟩ + δv. (4.61)

Through multiple repetitions of obtaining M, it is thus possible to repeat and average to

abate the effect of reduced measurement fidelity and fully calibrate the joint readout. As for
crosstalk, the joint readout employs a single physical line, and to some degree the crosstalk
between the states of the two-qubits is built into the measurement operator with the β12 term.
We call the ratio of this to the single-qubit polarizations, β12 /β1,2 the jointness, reflecting
the two-qubit state sensitivity of the measurement operator. Therefore, with a joint readout,
instead of striving to reduce the crosstalk, it is simply an inherent part of the measurement
to be calibrated and taken advantage of for full state detection.

4.4.3 Entanglement by joint measurement

A further facet of the joint readout is the ability to induce conditional entanglement through
the act of measurement. This situation arises in the special case when the cavity pulls of two
qubits are equal, such that χ(1) = χ(2) = χ (figure 4.8). In this case, the dispersive Hamiltonian
can be writeen as
1 ( j)
H = [ωC + χ (σz + σz )] a† a + ∑ (ω( j) + χ)σz + χ (σ+ σ− + σ− σ+ ) .
(1) (2) (1) (2) (1) (2)

j=1,2 2

(4.62)

This Hamiltonian can allow a QND measurement of (σz + σz ), as it commutes with

(1) (2)

the Hamiltonian. Since the states ∣01⟩ and ∣10⟩ will have the same cavity pull, the cavity is in
fact un-shifted from its bare frequency when the system is in either state and the either state
cannot be distinguished from the other through a measurement of (σz + σz ).
(1) (2)

This degeneracy in the measurment signal in regards to the state ∣0, 1⟩ or ∣1, 0⟩ can be a
way of generating entangled Bell states. The recipe is to first create a maximally superposed
state, [(∣⟩1 + ∣e⟩1 ) ⊗ (∣⟩2 + ∣e⟩2 )]/2 with R x (π/2) ⊗ R x (π/2), and then to perform the
(1) (2)

measurement by interrogating at the bare cavity resonance, corresponding to {∣0, 1⟩ , ∣1, 0⟩}.
The measurement operator would then be a projector on the Bell states, ∣ψ± ⟩ = (∣0, 0⟩ ±
√
∣1, 1⟩)/ 2,

M = ∣ψ+ ⟩ ⟨ψ+ ∣ + ∣ψ− ⟩ ⟨ψ− ∣ . (4.63)

104 cqed: qip with a photon bus

1.0 χ

∣, ⟩ ∣, ⟩
Transmission

∣, ⟩ ∣, ⟩
ωC
0.0
6.96 6.98 7.00 7.02 7.04
Frequency (GHz)
Figure 4.8: Dispersive peaks for generating Bell states by measurement. When the two qubits
have equal dispersive shifts χ, the cavity transmission peaks corresponding to ∣01⟩ and ∣10⟩
overlap. Then a measurement tone on this cavity peak would project onto a Bell state.

The act of measurement would leave the maximally superposed state in a Bell state . This
is generally known as entanglement generation conditioned upon the measurement of no
cavity-pull.

4.5 Chapter summary

In this chapter, we have developed some of the basic ideas for quantum information processing
in a circuit QED system. Microwave frequency pulses and fast flux bias pulses can be used
to perform single-qubit rotations. Furthermore, the interaction with the photon bus allows
for a number of two-qubit interactions which we may use to generate an entangling gate.
√
Specifically, by using the virtual photon interaction, we can realize the iSWAP, and by using
higher-excitation interactions of the transmon, we can realize a c-Phase gate. Furthermore,
we have introduced a joint qubit readout scheme built into the circuit QED architecture. We
will need to experimentally verify and calibrate this measurement model and use it for state
tomography. The remaining part of the thesis will detail our experimental implementation of
the features presented in this chapter.
 CHAPTER 5

Experimental Setup and Details

W
e now shift our focus towards experiments and lay some groundwork for the primary
results presented in this thesis. This chapter will give a brief discussion of the test
samples and hardware setup used for the experiments which will be described in detail in
the chapters to follow (chapters 6 to 9). This chapter will serve as a good background for
understanding how the different experiments came about and what specific investigations
could be performed.
First, we will identify the three experimental test samples (section 5.1) used for the
experiments. Then, we give a brief review of the basic fabrication techniques (section 5.2 and
section 5.3) involving optical and electron-beam lithography. We will also detail a number
of design considerations for the different transmon qubits tested in this work. Specifically,
we will discuss how we incorporate local flux-bias lines onto one of the samples for in-situ
fast qubit frequency tuning (section 5.3.3). Next, in section 5.4, the copper boxes and printed
circuit boards (PCBs) which shield, hold, and thermally anchor the transmon circuit QED
test samples are described, including introducing improvements which remove spurious
microwave resonances across the relevant bandwidth of our experiments. Then, we describe
the cryogenic circuitry for all the coaxial lines which allow us to address and readout our
samples in section 5.5. Finally, we review our room temperature control scheme (section 5.6

105
106 experimental setup and details

and section 5.7) and how we perform experiments both in the frequency domain and time
domain.

5.1 Experimental test samples

The results that are presented in this thesis in chapters 6 to 9 to follow are based on experiments
using three test samples, cQED187, cQED157, and cQED222. Though these three samples have
been the primary focus of the coupling qubits work in the past few years, many other samples
have passed through without achieving similar glory. In those samples, qubit frequencies
might not have been in the appropriate range for the experiments presented here, or they
might have been sacrificed over the course of incorporating the flux-bias line architecture.
Nonetheless, the description about the fabrication process will be presented with regards to
the development of the three specific samples.
cQED187 was used for the vacuum Rabi and single-qubit benchmarking experiments
(chapter 6). cQED157 was used for demonstrating a cavity bus and virtual swap interaction
with two qubits (chapter 7). cQED222 was used for generation and joint detection of highly
entangled qubit states, violation of a Bell inequality (chapter 8), and demonstration of two-
qubit algorithms (chapter 9).
In all cases the overall fabrication methods, from optical to electron-beam lithography,
are relatively straightforward, with the only major differences in the actual designs of the
transmon qubits germane to the different experiments tested.

5.2 Resonator Fabrication

A more detailed description of the resonator fabrication techniques and characterization

is given in Ref. [100] and Ref. [53]. Here in this section, we will focus on the optical design
considerations and standard optical lithography protocols in regards to the three samples used
for the experiments presented in this thesis. This stage of the fabrication process benefitted
enormously from working together with Luigi Frunzio and graduate student Joseph Schreier.

5.2.1 Resonator parameters

The resonators in all three samples are of the coplanar waveguide (CPW) geometry (figure 5.1)
defined using optical lithography. The resonator frequencies are defined by the length l of
 5.2. resonator fabrication 107

C in Cout b

l a t

top view cross section view

Figure 5.1: Coplanar waveguide geometry. The frequency of the CPW resonator is defined by
the length of the center pin between the gap capacitors at each end. The center pin width is
given by a and the gap between the center pin and ground planes is b. The ground planes are
designed to be much wider than b in order to be effectively infinite.

the center conductor and the dielectric єeff of the underlying substrate and the vacuum above
the chip, via the formula [101]
cπ
ωC = √ . (5.1)
l єeff

The CPW, with center pin width a, gap between center pin and ground planes b, is
designed based to have a characteristic impedance Z0 = 50 Ω. Although there are analytical
expressions which give Z0 for the CPW geometry [101], the determination of the designed a
and b is aided by the microwave simulation software TXLINE, part of the AWR Microwave
Office package. The quality factor is determined by the size of the gap coupling capacitors,
π 1
Q= , (5.2)
2 ω λ/2 Z0 (Cin
2 2 2
+ Cout
2
)

where ω λ/2 is the frequency of the fundamental λ/2 resonance. The CPWs can be defined to
be either symmetric Cin = Cout or asymmetric Cin ≠ Cout . Asymmetric coupling capacitors
can be used for increased collection efficiency, as a stronger output coupling capacitor would
give the microwave photons a preferred path for leaving the cavity. cQED157 and cQED187
are both asymmetric cavities, with a Q dominated by the output capacitor, whereas cQED222
is a symmetric cavity.
108 experimental setup and details

 µm
(a)

(b)

Figure 5.2: Optical images of resonator topologies. (a) Two-port resonator device as used
for cQED157 and cQED187. Two transmons are defined on the same side of the center pin,
but located at opposite ends near the input and output coupling capacitors. (b) Four-port
resonator device as used for cQED222. The two additional ports which enter from the top and
bottom sides are for the on-chip flux-bias lines. The ground plane is broken up into 4 distinct
pieces. The two transmon qubits are now located on opposite sides of the center pin, and still
on opposite ends of the cavity near the input and output coupling capacitors.

5.2.2 Optical lithography

cQED187 and cQED222 are defined using niobium on corundum (sapphire). The corundum
is 430 µm single crystal R-plane and a 200 nm thick layer of niobium is dc-magnetron
sputtered. After coating with S1808 resist, an optical mask with UV exposure defines the
coplanar waveguide for an etch. Reactive-ion etching is performed using SF6 as the primary
agent, followed by liftoff in acetone. The resonator frequencies for both samples are designed
to nominally give 7 GHz λ/2 resonance. The quality factors for cQED187 and cQED222 are
designed to be 20,000 and 10,000, respectively. cQED187 employs a single finger capacitor
on each of the input and output ports, but with different finger lengths of 30 µm and 100 µm.
These designs give rise to simulated capacitance values of about 2 fF and 5 fF, respectively.
cQED222 uses the same 100 µm finger, but on both input and output ports.
 5.3. transmon fabrication 109

cQED157 is made out of aluminum on silicon dioxide/silicon. The substrate is 500 µm

thick silicon with a thermally grown cap layer of 220 nm thick silicon dioxide. After coating
with LOR5A and S1808 resist, an optical mask with UV exposure defines the CPW for a
positive deposition step. 180 nm of aluminum is deposited followed by liftoff in NMP. The
λ/2 resonator frequency is designed with a center frequency of 6 GHz. The quality factor is
designed to be ∼ 300, with asymmetric input and output capacitors given by approximately
4 fF for a single finger design and 54 fF for an eight finger design.
A critical change in the design of the samples is the switch from a 6 GHz resonator
(cQED157) with a Q = 300 to a 7 GHz resonator with a Q = 10, 000 ∼ 20, 000 (cQED187 and
cQED222). Chronologically, cQED187 and cQED222 were fabricated after cQED157, reflecting
our new understanding with respect to the multi-mode Purcell effect (section 3.5.1). We
changed the design to push the dip in the real part of the admittance between the fundamental
and first harmonic frequencies higher up and away from the band in which we operate our
qubits (typically ∼ 4 − 8GHz).
Also note the difference in the ground plane for cQED222 in figure 5.2, permitting
additional CPW traces for the on-chip flux bias controls. Incorporating these flux bias lines
breaks the ground plane into 4 distinct pieces rather than 2. As we will discuss later in
this chapter in section 5.3.3, on-chip wire bonds or air-bridges can be used to connect all
of the ground planes into a single piece. This will play a role in the coupling of the qubits
to non-CPW modes on the chip, which otherwise would adversely affect qubit coherence
properties.

5.3 Transmon fabrication

All three samples are fabricated with two transmons on each, located on opposite ends of the
cavity, near the input and output capacitors. The two transmons on each sample are defined
in a single electron-beam lithography step. A bilayer resist system is used, with a top 100 nm
thick layer of 950K PMMA A3, on top of a copolymer 550 nm thick layer of MMA(8.5)-MAA
EL13. The bottom layer is more sensitive to electrons than the top, resulting in a natural
undercut of ∼ 80 nm during electron beam writing. The undercut is a necessary feature for
achieving good clean liftoff with aluminum. The full lithography recipes follow those outined
in Ref. [53] and were performed with Luigi Frunzio, graduate student Blake Johnson, and
postdoc Leonardo DiCarlo.
110 experimental setup and details

(a)  µm (c)

 µm

(b)  µm (d)

 µm

Figure 5.3: Optical images of different transmon designs. (a) Standard transmon design
employed in cQED157 and on one of the qubits in cQED187. (b) Balanced transmon design
used in one of the qubits in cQED187. (c) and (d) Transmon designs incorporating flux bias
lines. A slightly different transmon SQUID loop design is necessary to accommodate the
flux bias lines entering from the (c) bottom of the chip or from the (d) top of the chip, while
preserving the same double-angle evaporation procedure.

For each sample, the junctions in both transmon qubits are designed in a single step. The
Dolan bridge technique which is described in Ref. [53] is used for fabricating the junctions.
This method is especially useful for making small junctions, as for the transmon qubits used
here, the areas are typically 150 nm by 250 nm.
The similarities between the three samples ends at the level of junction fabrication however.
The sizes of the SQUID loops and the values of the relevant capacitances are important
design considerations for each of the test samples, and permit them to be used for specific
sets of experiment. The most traditional transmon (section 5.3.1) design is described in
detail in Ref. [53]. However, here we present two additional designs, the balanced transmon
(section 5.3.2) and the flux-bias transmon (section 5.3.3).

5.3.1 ‘Traditional’ transmon

A cartoon of all of the relevant pieces of metal which must be considered for the transmon
design is shown here in figure 5.4. The charging energy EC and the voltage division ratio β
are defined in terms of all of the capacitances between pieces of the circuit figure 5.4. With
regards to describing the fabrication and design considerations, we can be more specific
about which of the different pieces in the circuit of figure 5.4 affect the transmon and cavity
 5.3. transmon fabrication 111

Figure 5.4: The capacitance network for the transmon in a coplanar waveguide resonator. a
The complete circuit diagram, showing all the capacitances, designed and parasitic, between
the 5 metallic areas of the transmon-cavity circuit, shown in b (not to scale). c The simplified
equivalent circuit which can be found by using the electrical engineer’s rules for series and
parallel capacitors. (Figure used with permission from [61]. See Copyright Permissions.)

coupling parameters. Piece 1 refers to the lower ground plane (yellow); piece 4 is the CPW
center pin (red); piece 5 is the upper ground plane (green); piece 2 is the lower transmon
island (cyan); piece 3 is the upper transmon island (blue).
All samples employ a pocket size of 300 µm by 30 µm cut out of the lower CPW ground
plane, in which the transmon islands can be defined. In the traditional transmon design,
as shown in figure 5.3a, we get typically EC = 300 ∼ 400 MHz with β ∼ 0.10 − 12, which
yields 01 /π = 150 ∼ 250 MHz. The best way to adjust EC is through changing the capacitance
between the islands, C23 . The coupling capacitance C  , is primarily determined by the length
of piece 3, typically 290 − 300 µm, and the gap between piece 3 and piece 4, typically 3 − 5 µm.
The standard transmon design gives simulated values of C34 ≈ C23 = 24 fF. SQUID loops are
typically designed with dimensions in the range of 2 − 3 µm × 2 − 3 µm.
112 experimental setup and details

5.3.2 Balanced design

As a result of the location of the transmon between the center pin and one of the ground planes
of the CPW, it is possible for the transmon to couple to a slotline mode [102] of the resonator
where the top and bottom ground planes oscillate out of phase. The balanced transmon is
the result of an attempt to balance the asymmetry between the top and the bottom line using
a Wheatstone bridge technique (to be discussed in more detail in Ref. [103]). It was proposed
[61] that coupling to such a slotline mode could cause a Purcell-like spontaneous decay, and
lead to shorter qubit relaxation times.
Although the balanced design was originally intended to make the effective coupling to a
slotline mode minimal, one of the byproducts of the design was actually a much smaller 01
due to a reduced voltage division ratio β ∼ 0.08. Whereas most traditional transmons have
coupling strengths in the 200 − 300 MHz range, these balanced transmons are on the order
of 50 − 100 MHz. Figure 5.3b shows one such balanced design, employed in cQED187. The
diminished coupling is primarily due to the reduced length of piece 3, resulting in a smaller
direct capacitance to the center pin C34 . The balancing is achieved through increasing C31 ,
C24 , and C25 .
In the experiments presented in this thesis, the intended effect of the balancing in regards
to slotline mode coupling is not tested. However, the smaller 01 side-effect is critical for the
strongly-driven vacuum Rabi experiments, permitting us to see higher order photon interac-
tions in the Jaynes-Cummings ladder (section 6.1.1) for characterizing our state initialization
(section 6.1).

5.3.3 Flux-bias transmon design

cQED222 employs a different transmon style which permits on-chip flux tunability. The design
of not only the qubit (two designs shown in figure 5.3c–d), but the entire chip (figure 5.2c)
is modified. The transmon SQUID loop and flux-bias line traces are designed to achieve
enough coupling for sufficient qubit transition frequency tuning.

Flux coupling

Together with former postdoc Johannes Majer, we designed the flux-bias line (FBL) to be
itself a 50 Ω coplanar waveguide which runs perpendicular to the CPW resonator on the
chip towards the transmon pocket. At the pocket, the FBL is terminated in a short circuit to
 5.3. transmon fabrication 113

the ground planes. An optical zoom-in image on the flux-bias line termination is shown in
figure 5.5a. The CPW which feeds the FBL tapers down to 2 µm before being short-circuited
near the pocket for the transmon with a width of 2 µm and length of 10 µm on each side. By
placing the loop off center, as shown in the optical image figure 5.5a, and right up against
the split inductive short permits one arm of the inductive short to couple more strongly
than the other, such that a non-zero flux is threaded through the loop. Although the mutual
inductance between the FBL and the loop can be calculated from simple electrodynamics
and simulations, the presence of superconductor around the actual FBL and qubit loop can
result in flux focusing via the Meissner effect, which significantly increases the magnetic field
coupling by a factor of 2-3. From initial experiments on FBL qubits, we have found it takes
∼ 6−10 mA at the sample to tune half a Φ0 for SQUID loop sizes of 8−10 µm ×5−7 µm. With
regards to the entire FBL circuitry up to room temperature (∼ 300 K), the amount of current
applied to the external line will be determined by the total attenuation and filtering of the
line used for thermalization (section 5.5). Therefore, it is also important to pay attention to
the thermal heat load on the line to not swamp the cooling power of the cryostat. In practice,
transmon tunability using a current range at the sample of ±10 mA can be afforded without
significantly affecting the base temperature (+ ∼ 5 − 10 mK).

Relaxation of the flux-bias transmon

While the flux-bias lines provide enough magnetic coupling to sufficiently tune the qubit
transition energies, it is an additional connection which can also electrostatically connect
the qubit to the 50 Ω environment. As discussed in section 3.5.1, the key quantity to consider
for qubit relaxation is the real part of the total impedance seen by the qubit. In the standard
transmon design, the impedance which we find to be the primary culprit is that of the
transmission line resonator, resulting in the multi-mode Purcell effect.
However, separate characterization experiments that first implemented FBL transmons
demonstrated qubit relaxation rates that did not follow just the simple multi-mode Purcell
effect [82] compounded with the intrinsic Q ∼ 70, 000. Instead, in these samples, which had
a cavity frequency of 7 GHz, measured relaxation times at frequencies above 7 GHz would
fall off very precipitously and at much lower frequencies than would be predicted due to the
λ mode of the transmission line cavity. As a result, together with postdoc Leonardo DiCarlo,
we moved towards performing simple simulations based on the FBLs using both Microwave
Office and a high frequency electromagnetic simulation software named Sonnet.
114 experimental setup and details

(a) (b)

Cf
Lf Z

Figure 5.5: The flux-bias line. (a) Optical image of the flux-bias line short-circuited termination
near the SQUID loop of the transmon qubit. (b) Modeling the flux-bias line as a circuit-
decomposition, with a capacitance in series C f with the transmon, and a shunt inductance L f .
The inductance is accounted for by the 2 µm by 20 µm short-circuited termination to ground.

For the Microwave Office simulations, we can model the circuit with the FBLs using a
capacitance C f in series with the transmon and a shunt inductance L f , as shown in figure 5.5c.
The capacitance can be estimated to be ∼ 3 fF and the inductance to be ∼ 20 pH (from
electrostatic simulations). For this simple model, the real part of the admittance can be found
to be
1 ω 4
Re[Y] = ( ) , (5.3)
Z0 ω0
√
where ω0 = 1/ L f C f . This reflects a much steeper fall-off of the T1 with respect to frequency
than is generally the case without the FBLs. However, it does not explain the sharp drop off
immediately above the fundamental CPW mode.
Sonnet provides a different method of simulation by giving high frequency electromag-
netic field calculations based on 2D geometries. The full resonator with two FBLs can actually
be designed in the software and the real part of the admittance seen from the location of the
qubit, labeled port 1 in figure 5.6, can be determined. Assuming a qubit coupling capacitance
of C  = 15 fF, a relaxation time, or T1 , curve due to the entire flux bias line coplanar wave-
guide resonator is found and plotted in figure 5.7. The overall asymptotic behavior agrees
with the simple 1/ω4 model from the simple Microwave Office simulation. However, the
Sonnet simulations tell us a lot more information about the resonant structure of the full chip.
 5.3. transmon fabrication 115

2 1 4

Figure 5.6: FBL schematic for Sonnet simulations. For performing the Sonnet simulations, a
simplified chip and transmon design is used. We label 5 ports, and find the total admittance
referenced to port 1, where the transmon qubit is located.

Specifically, besides the presence of the two resonances due to the λ/2 and λ CPW modes at
f0 = 6.45 GHz and f1 = 12.8 GHz, a third mode is found at fs = 11.65 GHz.
The location of this third mode in between the two standard CPW modes agrees qualita-
tively with the experimental drop off in the relaxation times at frequencies directly above the
λ/2 resonance. We can identify this mode with a slotline, or ‘wiggle-waggle,’ mode which is
due to an odd transmission line mode between the lower and upper ground planes of the
CPW. This is the same mode to which the modified transmon balanced design (section 5.3.2)
attempts to decrease the coupling to. In the standard CPW with transmon designs, the
presence of this mode has not been detected in T1 versus transition frequency measurements.
However, in the case of the CPW with flux bias line transmons, the flux bias lines provide a
path of coupling the environment to such a mode. This is best seen by using Sonnet to view
the current density over the entire chip as a function of frequency, as shown in figure 5.8.
We find that at the fundamental and first harmonic frequencies of the CPW resonator, the
current density is primarily distributed over the center pin and concentrated around the gap
capacitors which define the cavity. This reflects a preferred loss path for photons in the cavity
116 experimental setup and details

0
10 with flux bias lines
without flux bias lines
-1 simple model
10

-2
10

-3
10
T (s)

-4
10

-5
10

-6
10

-7
10
2 4 6 8 10 12 14
Frequency (GHz)
Figure 5.7: T1 Simulations of resonators with and without FBLs. The simulation software
returns the Re[Y], from which we calculate T1 . Without the presence of the FBLs, we find the
standard multi-mode Purcell curve (magenta squares), with two dips corresponding to the λ/2
and λ CPW resonances. With the presence of the FBLs (blue circles), we see an extra dip, at
11.65 GHz. The overall behavior of the FBL case also agrees over the frequency range according
to the simple 1/ω4 model.

out through the capacitors, which is the expected Purcell effect for spontaneous emission
(section 3.5.1). However, at fs corresponding to the wiggle-waggle mode, current becomes
concentrated over the flux bias line ports, resulting in a different path for spontaneous emis-
sion. Therefore, whereas the wiggle-waggle mode may not have been a serious problem for
previous transmon designs, for the flux-bias line sample it poses a significant obstacle to
operating with qubit transition frequencies above the fundamental CPW mode.

On-chip wirebonds

It is possible to employ some on-chip engineering to abate the issue with the wiggle-waggle
mode. Specifically, by connecting the upper and lower ground planes via an air-bridge in the
middle of the center-pin line, as shown in figure 5.9a, we can actually move the frequency of
 5.3. transmon fabrication 117

6.45 GHz

λ/2 resonance

11.65 GHz

wiggle-waggle

12.8 GHz

λ resonance

Figure 5.8: Current density simulations of resonators with FBLs. Color scale shows the
current density across the chip. Blue represents no current and red represents high current. For
the λ/2 and λ resonances, we can see the radiation along the center pin and, coupled through
only the input and output capacitances on each end. However, for the wiggle-waggle resonance,
there are high current paths along the flux-bias lines and all along the center-pin.
118 experimental setup and details

(a)

1
3
-3
10
(b)
-4
10

-5
10
T 1 [sec]

-6
10

-7
10 no bondwire
L = 600 µm
-8 L = 400 µm
10
L = 200 µm
-9 L = 100 µm
10
6 8 10 12 14 16 18 20
Frequency [GHz]
Figure 5.9: Pushing up the wiggle-waggle with bondwires. (a) Schematic for Sonnet simula-
tion incorporating and on-chip air-bridge for connecting the upper and lower ground planes
at the center of the CPW. L is the length of such a bridge. (b) Simulated T1 curves for resonator
with FBLs and on-chip air-bridge with varying L from 600 µm to 100 µm. With decreasing L,
the wiggle-waggle resonance is pushed up in frequency.
 5.3. transmon fabrication 119

 µm

Figure 5.10: Optical image of an on-chip wirebond. Wirebonds across the center pin are
generally 200 − 300 µm long. In standard practice, we place 3 wirebonds in parallel.

the wiggle-waggle mode. By including such an airbridge in our Sonnet simulations, we find
that the wiggle-waggle mode can be pushed past the λ CPW resonance and away from the
standard qubit operating regime. Figure 5.9b shows that varying the distance L between the
location of the air-bridge and the center pin changes where the position in frequency of the
wiggle-waggle resonance.
In experimental practice, this air-bridge can be achieved by placing on-chip wirebonds
which go over the center pin but connect the lower and upper ground planes. Figure 5.10
shows an optical image where one such wirebond is placed across the center pin in the center
of the CPW line. We can perform simple 4 K transmission experiments with Nb resonators to
demonstrate the effect of the wirebonding. Figure 5.11a shows the transmission measurements
of a 7 GHz Nb resonator, with a Q = 100, and having performed no modifications to the
chip or sample holder (described in section 5.4). S21 reflects the standard input-output CPW
transmission path. S2L (S2R ) reflects applying the drive to the left (right) flux-bias line port
and transmitting out the output path.
Besides the λ/2 and λ resonances, a peak in the transmission in all three measurements is
seen around 11 GHz. By adding on-chip wirebonds at the center of the sample and repeating
the measurements, we find the transmission spectrum given by figure 5.11b, where the struc-
120 experimental setup and details

0
(a) S
R

Insertion Loss (dB)

-20 SL
SR
1 2 -40

L -60

-80
0
(b)
R

Insertion Loss (dB)

-20

1 2 -40

-60
L
-80
6 8 10 12 14
Frequency (GHz)

Figure 5.11: Experiment showing transmission spectrum with and without wirebond.
(a) Transmission experiments are performed in a 4 K dunk test set-up. We measure the
standard CPW transmission S21 (blue), the transmission from the left FBL to the output S2L
(red), and the transmission from the right FBL to the output S2R . The black dashed lines
indicate the location of the CPW λ/2 and λ resonances. The wiggle-waggle mode can be seen
at around 11 GHz. (b) After re-dunking having added on-chip wirebonds to the center of the
chip, the spectrum is improved considerably, with the wiggle-waggle resonance previously
seen in S2L and S2R now pushed up near 13 GHz.

ture at 11 GHz is no longer there. Other resonances are due to copper traces on the sample
board holder as well as the sample box, which will be the subject of the next section.
The flux-bias line sample studied in this thesis, cQED222, employs such a set of three
wirebonds over the center pin to suppress this wiggle-waggle mode and all flux-bias line
samples now employ this same technique.

5.4 Sample boards and holders

Having described the samples at the chip-level, we now proceed up through the experimental
setup onto the sample boards and sample boxes. We need both a sample holder and box
which suppress parasitic resonances and can be easy to use and convert into microwave
coaxial cable lines for input and output through the cryostat. cQED187 and cQED157 each
have only two ports each, and we use a simple design known as the ‘coffin’ class (section 5.4.1)
 5.4. sample boards and holders 121

of sample holders. However, cQED222 is a bit more complicated, with 4 ports, an input,
output, and two connections for flux bias line control. In that case we employ the ‘octobox’
class (section 5.4.2) of sample holders.

5.4.1 Coffin design

The coffin design consists of two ports, an input and an output, for drive and readout lines to
a CPW resonator. The printed circuit board (PCB) is copper plated on the top and bottom of
an Arlon AD1000 (єr = 10) dielectric with a single milled out slot to the bottom copper plane.
There are locations for two Rosenberger SMP connectors for the input and output ports. A
50 Ω CPW line is etched into the copper, with a center pin width of 150 µm and gap to ground
plane of 90 µm. The upper and bottom ground planes are connected by copper coated vias
(12 mil hole size) which are drilled throughout. These vias reduce the parasitic modes which
can arise due to imbalances in the top and bottom copper ground planes. A 2 mm × 7 mm
chip can be mounted into the slot. Many wirebonds are used to attach ground planes of PCB
to ground planes of chip and three wirebonds are used for connecting the ground planes
across the center of the chip. A picture of the coffin PCB is shown in figure 5.12a.
The coffin PCB is mounted onto a copper plate which is then covered by a rectangular
copper box, with two output ports which mate to an SMA terminated coaxial cable. Rosen-
berger SMP connectors are soldered onto the slots of the board, and SMP bullets connect the
board to the SMA terminated coaxial cable. The full set-up is shown in figure 5.12b.

5.4.2 Octobox design

The octobox design, shown in figure 5.13, expands the number of connections from 2 to 8,
allowing for geometries with two chips, each with an input and output, or for a single chip
with multiple connections for extra control lines of multiple qubits on a sample. In the case
of cQED187, the octobox is used, but for the simplest experiment of just two lines, with an
input and an output. For cQED222, we use 4 ports of the octobox, input, output, and a line
for each qubit’s fast flux bias line.
Whereas the coffin design has a relatively clean microwave spectrum between 2 to 12 GHz,
the octobox and its corresponding PCB actually suffer from quite a number of resonances.
Figure 5.14a shows the transmission of a simple bare 7 GHz resonator with on-chip flux-
bias lines and on-chip wirebonds as discussed earlier, mounted on an octobox PCB with
8 connections to the single chip and placed in the copper octobox. Although the on-chip
122 experimental setup and details

(a)

(b)

Figure 5.12: Coffin box holder. (a) The sample is placed into a 2 mm by 7 mm pocket located
in the center of the PCB. The PCB connects to semirigid coasxial cables via SMP surface mout
connectors using a bullet. Vias are drilled into the PCB and copper plated to connect the top
and bottom ground planes. (b) Inset shows the entire rectangular coffin box with SMA to SMP
jumpers.

wirebonds removed the structure at 11 GHz, there are still many other resonances throughout,
including a very strong one at 10 GHz. This resonance corresponds to a whispering gallery
mode of the box, removable by filling up the three-dimensional cavity within the sample
holder.
A lot of the structure is actually removed from the system by modifying the octobox
with a flip-chip combined with a larger copper back plate. The flip-chip is simply another
PCB, or thin copper plate, with the appropriate holes cut into it sitting on top of the standard
PCB, effectively placing another ground plane right on top of the CPW traces, as shown in
figure 5.13. This makes the geometry more symmetric and limits any modes which may arise
due to differences in the dielectric on top and below the copper traces. The larger copper back
plate serves to fill up up most of the volume within the box, removing whispering gallery
and other 3D resonator modes. Together, these effectively eliminate any parasitic box modes,
 5.5. cryogenic setup 123

Figure 5.13: Octobox holder. The far left is the copper octobox which permits 8 connections to
a sample. This particular copper octobox is designed to fill up the entire space in between the
box and the sample lid. The piece in the middle is the “flip-chip” which is similar in size to a
PCB, but made out of copper with traces milled out to avoid shorting to transmission lines on
the PCB. The piece on the right is the lid with 8-port PCB but only a two-port sample attached.

and the difference is night and day as shown in the transmission spectrum before and after
the changes of figure 5.14b.
The development of this flip-chip and reduced mode volume is now incorporated into all
of our experiments.

5.5 Cryogenic setup

The experiments were performed using a Cryoconcept 200µW@100 mK dilution cryostat,

capable of attaining base temperatures 10 − 15 mK, with the samples at temperatures 15 ∼
20 mK. The key details of the cryogenic setup pertain to the reduction of heat and noise
that can reach the sample from room temperature or microwave cryogenic amplifiers. A
schematic of the cryogenic circuitry is shown in figure 5.15.
For samples cQED187 and cQED157, the only pathways to the sample are the RF drive and
the RF output. Johnson noise from room temperature on the RF drive line is attenuated with
20 dB at the 4 K stage, followed by 30 dB at the 15-20 mK base temperature plate (figure 5.15).
The long cryogenic lines give an additional ∼ 10 dB resulting in a total line attenuation of 60 dB.
We can permit this level of attenuation and still be able to perform on-resonant measurement
124 experimental setup and details

1 2

L
0
(a) S
Insertion Loss (dB)

-2 0 SL
SR
-4 0

-6 0

-8 0
0
(b)
Insertion Loss (dB)

-2 0

-4 0

-6 0

-8 0
6 8 10 12 14
Frequency (GHz)
Figure 5.14: Resonator with FBL in octobox. With additional wirebonds, the transmission
spectrum is further improved to (a). However, there is still a strong resonance in all 3 measure-
ments at ∼ 10 GHz, corresponding to a 3 dimensional cavity mode. This is suppressed with the
use of the flip-chip and filled octobox design, with which we measure the transmission spectra
shown in (b).
 5.5. cryogenic setup 125

of the cavity (∼ −80 dBm) as well as off-resonant drive of the qubits (∼ −20 dBm), without
significantly heating the attenuators at the cold stage. All attenuators in the cryogenic samples
are made by XMA.
On the RF output line, the primary noise source is in fact the cryogenic amplifier at 4 K.
These amplifiers have typical noise temperatures TN = 5 K. It is not advisable here to use
attenuation between the output port of the cavity and the amplifier as the whole point of the
amplifier is to increase the amount of signal. However, a microwave circulator allows for the
signal to pass through to the amplifier without being attenuated, while taking all the reflected
noise off of the amplifier and dumping it in a 50 Ω termination instead of reaching the sample.
In our experiments, we used high-electron mobility transistor (HEMT) amplifiers, made by
Caltech Radiometer Group, Model Numbers LNA93D and LNA95D, with low temperature
gain of ∼ 33 − 36 dB.
The location of the microwave circulator can affect the mean photon number in the
cavity. For cQED187, experiments were performed in two cryogenic configurations, with
two circulators at the 100 mK plate of the cryostat, and with an additional third circulator
at the 20 mK base plate. As we will show in section 6.1.1, in fact, this simple change means
a difference in the number of photons going from 0.018 down to 0.003. The mean photon
number was extracted using fits to strongly driven vacuum Rabi spectra [89]. For cQED157,
the experiments were actually performed before adding the third circulator. Then, finally
for cQED222, there were two broadband (4–12 GHz isolators (circulators with built-in 50 Ω
terminations, Pamtech Model No. CWJ1019), thermally anchored to the base temperature
plate.
The flux bias line sample cQED222 actually requires a bit more of care in terms of the
cryogenic thermalization, because in addition to the RF drive and output, there are two
additional lines for each qubit flux-bias. The primary requirements of such a line are the
ability to tune through at least a single flux quantum on the SQUID of each qubit as well as
the ability to allow enough bandwidth for tuning of the qubits on fast nanosecond timescales.
The flux bias lines are first attenuated with 20 dB at 4 K. Then at base temperature, the lines
pass through first a Mini-Circuits VLFX-1050, 1 GHz low-pass filter, followed by a “chocolate”
powder or eccosorb powder filter (figure 5.15). These dissipative powder filters have the
characteristic of allowing through DC-300 MHz, followed by a sharp exponential roll-off of
the transmission for higher frequencies. The 20 dB at 4 K is chosen to afford enough current
for biasing the transmon across a single Φ0 . The technique of using the powder filters for
126 experimental setup and details

cQED222
only

20 dB

20 dB
R.T.
UT-85-SS
HEMT
30 dB

20 dB

20 dB
4K

UT-85-SS/SS
15 mK
m-strip VLFX-1050

m-strip VLFX-1050
30 dB

circulators

lossy M-C

lossy M-C

cavity cavity left right

in out flux bias flux bias
Figure 5.15: Schematic of cryogenic circuitry. There are a total of four cryogenic lines: input,
output, left flux-bias and right flux-bias. The experiments for cQED157 and cQED187 do not
employ the FBLs.
 5.6. room temperature control 127

thermalization is very analogous to that used for DC gate control lines in Cooper-pair box
circuit QED experiments [53].

5.6 Room temperature control

Outside of the cryostat, we have all of the control components which allow us to apply
microwave signals to address the cavity and the qubits, as well as the components necessary
for readout of the output line. For the cavity, an Agilent microwave signal generator E8257D
is used to address either the 5 GHz or 7 GHz cavity. A built-in pulse modulation feature
allows for shaping of measurement pulses with a bandwidth of ∼ 10 MHz. The qubits are also
addressed using Agilent microwave signal generators. Normally, the qubit frequencies are
detuned from the cavity frequency, and hence the qubit drives are off-resonant and need to be
higher in power. All of the signals are added using microwave power splitters (Mini-Circuits
ZFSC-2-10G) used in reverse. Pulse-shaping of the qubit signals is done using IQ modulation,
which is another feature built into Agilent IQ microwave generators E8267C/E8267D. A
detailed schematic is given in figure 5.16.
For cQED187 and cQED157, the qubit transition frequencies are tuned via an external flux
produced by the superconducting coil within the body of the Cryoconcept dilution cryostat.
This superconducting coil produces 335 gauss per ampere. By biasing either a 10 kΩ or 1 kΩ
resistor with a Yokogawa DC voltage source, we are able to tune across many Φ0 for each
qubit.
cQED222, with its built-in on-chip FBLs, obviates the use of an external magnetic field.
Instead, each qubit FBL is connected to its own Yokogawa voltage source, driving a 20 dB
attenuator at room temperature, for DC tuning. For fast tuning (up to bandwidth of ∼
300 MHz, each FBL is connected to a channel of a Tektronix AWG5014 arbitrary waveform
generator (figure 5.16).
The output line is further amplified outside of the cryostat with a pair of amplifiers, Miteq
ULN-10 and ULN-35, with quoted max noise figures of F = 1 and F = 3.5 and a gain of 33
and 23 dB, respectively, over a bandwidth of 3 to 8 GHz.
Since the state of the qubits are encoded in the phase and amplitude of the transmitted
cavity signal, we can use an IQ demodulation technique in either homodyne or heterodyne
[53]. In the heterodyne detection scheme, an IQ mixer (Marki Microwave IQ0307MXP) is
used to mix down, as a demodulator, such that the output signal enters through the RF port,
and a microwave tone which is 1 ∼ 10 MHz detuned from the cavity signal is applied to the
128 experimental setup and details

VL
VR flux bias line controls

Tektronix
AWG 5014
IL
fL I Mini-circuits
ZFSC-2-10G
Q
Trig QL 1
S 1
Agilent 2 S
E8267C 2
E8267D input
IR
fR I
Tektronix
AWG 520

QR Q

fC
M 40 dB
marki
E8257D IQ0307MXP
Trig IQ Mixer
Vh,I fC I
Acqiris

SRS
AP240

LO RF
445A Q
Vh,Q E8254A output

Figure 5.16: Room temperature control schematic.

LO port. The IQ demodulator gives two outputs, one in-phase and one 90○ out-of phase at
an IF frequency equal to the detuning between the LO frequency and the applied drive at the
RF port. Homodyne detection refers to using an IF frequency of 0, such that the final signals
on both channels are simply DC and correspond to two signals that correspond to I and Q
quadratures of the cavity signal.
The two IF signals go through a final stage of low-bandwidth amplification using an
Stanford Research Systems 350 MHz (Model No. SR445A) preamplifier, before finally entering
two channels of a 1 GS/s Acqiris AP240 acquisition board. When performing heterodyne
detection, a further digital demodulation is performed known as digital homodyne. Here,
only a single output of the IQ demodulator is kept and a sine and cosine at the IF frequency
 5.7. pulse control and modulation 129

is digitally multiplied to the signal. The outputs are then a digital I and a digital Q, which
√
can then be combined to give an amplitude A = I 2 + Q 2 or a phase ϕ = tan−1 (I/Q). All
components are locked in phase via a common SRS 10 MHz rubidium frequency standard.

5.7 Pulse control and modulation

Qubit control pulse generation is performed via an Agilent vector IQ microwave frequency
generator (E8267C/E8267D). Although such a piece of hardware is not the only way of pro-
ducing good dual-quadrature control pulses, it is very reliable in terms of timing, bandwidth,
and linearity. To control each qubit, we are looking for microwave carrier frequencies in the
3 ∼ 10 GHz range, while shaped with pulses that have nano-second resolution, at a bandwidth
of 30 ∼ 300 MHz.
The pulses are programmed in either Labview or Mathematica, and then imported into
either a Tektronix 4-channel AWG5014 or 2-channel AWG520 arbitrary waveform generator.
Both generators provide at least 1 GS/s and a voltage amplitude of 2 V peak-to-peak for driving
the vector generator’s internal IQ mixer. The vector control of the pulses allows us to apply
signals either in-phase, or 90○ out-of-phase for rotations along the x and y directions of the
qubit’s Bloch sphere. Each quadrature corresponds uses up one channel of the AWG, such
that controlling the x and y rotations of a single-qubit requires two channels.
Current investigations are being performed to build a piece of hardware that takes a
single carrier frequency, splits it into two signals in quadrature, and mixes in a modulated
pulse shape. The resulting waveform is mixed down to the appropraite qubit frequency. This
technique of single-sideband modulation will allow us to remove reliance on the expensive
IQ generators provided by Agilent.

5.8 Chapter summary

In this chapter, we have reviewed the basics of circuit QED sample fabrication, and touched
upon the design considerations of the three samples investigated in this thesis. Overall,
the fabrication techniques are quite simple and reliable to within ∼ 10% of all parameters.
Currently the biggest variation which exists is still in hitting targeted EJ values. This is most
likely due to slight differences in the conditions during the electron-beam lithography step
for writing the Josephson junctions. Most of the other design parameters, including the
charging energy EC , cavity-qubit coupling , cavity frequency ωC , cavity quality factor Q, are
130 experimental setup and details

Table 5.1: Summary of measured sample parameters.

Qubit/Sample ID ωC /2π (GHz) κ/2π (MHz) /π EJmax /h (GHz) EC /h (MHz)

cQED157 qubit 1 5.22 33 105 14.9 420
cQED157 qubit 2 5.22 33 105 18.9 440
cQED187 qubit 1 6.920 0.3 347 18.0 400
cQED187 qubit 2 6.920 0.3 94.4 22.6 340
cQED222 qubit 1 6.902 1.0 199 28.48 320
cQED222 qubit 2 6.902 1.0 183 42.34 300

found in experiments to agree well with targeted specifications. Table 5.1 summarizes the
three samples discussed in this chapter in terms of experimentally measured parameters.
This chapter has also reviewed the experimental setups both inside of the cryostat and
outside. The cryogenic circuitry is very important for the system to behave as qubits in their
ground state coupled to a cavity. Next in the chapters to follow, we will investigate how we use
the room temperature control apparatus to perform basic aspects of quantum information
processing.
 CHAPTER 6

Initialization and Benchmarking of Single-Qubit Gates

T
he success of any computational architecture depends on the ability to perform a large
number of gates, and gate errors meeting a fault-tolerant threshold. The most advanced
classical computers today can perform up to 1015 operations without the need for error
correction. For a quantum computer, in order to maintain coherence throughout a long
string of operations, quantum error correction is a proposed necessity. Surprisingly, the
most conservative estimates place the required gate errors thresholds to be on the order of
10−4 [26, 104] for quantum error correcting codes to function. Yet, thus far such control of
quantum systems has been difficult to attain in experimental quantum systems.
For an experimental quantum computing system to be considered viable for quantum
error correction, the gate error rates must be characterized and understood. Here, we
benchmark the single qubit error rates for transmon qubits in a circuit QED system. Although
photons and trapped-ion systems remain the paragon for single qubit gate fidelity, reaching
upwards of 99.9%, solid-state systems are making rapid progress, and the full characterization
of single-qubit operations here demonstrate that the road ahead for superconducting qubits
is promising.
Of course to be able to perform single-qubit operations, it is critical for the qubit to
also start in a well-defined pure state. Although many qubit systems employ active cooling
techniques [105, 106] to initialize a ground state, most circuit QED architectures simply rely

131
 132 initialization and benchmarking

on external thermal cooling of the sample∗ . In this chapter, we will first present in section 6.1
a sensitive measurement of the cavity temperature to which the transmon qubits couple using
the nonlinear vacuum Rabi spectrum (as published in Ref. [89]), from which we can estimate
the polarization of the initial qubit state. Then, section 6.2 will discuss multiple techniques
for characterizing single-qubit operations in circuit QED, as based on Ref. [108]. Finally, we
will introduce some new qubit control techniques for further improving single-qubit gate
fidelities in the cavity coupled transmon system (section 6.4).

6.1 Initializing pure states

In circuit QED, where the qubit and cavity excitations are in the microwave frequency regime,
a pure ground state initialization of the qubit is strongly dependent on the thermal bath into
which the qubit decays. When the qubit is Purcell limited such that the primary decay channel
is via spontaneous emission through the cavity, the temperature of the cavity will determine
the equilibrium polarization of the qubit. The temperature of the cavity is directly reflected
as a thermal population of photons, as given in section 4.1 by (4.1). For our experiments
performed at cryogenic dilution refrigerator temperatures, with a base temperature of 20 mK,
this would ideally correspond to ⟨n⟩ ∼ 10−7 photons for a 7 GHz cavity.
Despite the nominally ∼ 20 mK base temperature, it is still important to be able to
experimentally verify the mean number of photons in the cavity, as the experimental setup
can result in elevated thermal noise and also non-equilibrium excitations. Although the
sample itself is thermally anchored to the base temperature, the control lines can still serve as
noise sources. For example, warm attenuators can be black body sources of radiation. Another
major source of noise is the cryogenic HEMT amplifier on the output port of the sample.
With a noise temperature of 5 K, a direct connection to the output port of the cavity would
be detrimental. As mentioned in section 5.5 however, we employ a microwave frequency
circulator in between the output port of the sample and the cryogenic amplifier to combat
this effect. Nonetheless, the reflected noise radiated from the amplifier is still dissipated in
a 50 Ω termination connected to the circulator, which could be a source of elevated mean
photon numbers in the cavity.
So what are some of the ways to experimentally detect the mean photon number in the
circuit QED system? One method is the ac Stark shift previously discussed in section 4.2.4.

∗
Active cooling may be helpful for the new fluxonium qubit design [107].
 6.1. initializing pure states 133

By directly increasing the number of photons in the cavity through a coherent drive, the
ac Stark effect (4.27), results in a linear shift of the qubit transition frequency (as long as
the ac Stark drive is not too strong). Another option is if the qubit and cavity are in the
number-splitting regime (section 3.4.3), then by driving the cavity with a coherent state and
observing the number-split spectrum of the qubit, it is possible to fit to a combined thermal
and Poisson-distributed spectrum, from which a mean photon number could be extracted.
Though both of these methods are sensitive to the level of ∼ 0.1 photon in the cavity [109, 110],
current signal-to-noise ratios in the detection of the qubit spectrum prevent them from being
sensitive to even lower photon numbers.
However, an interesting regime of circuit QED which actually provides a very sensitive
photon number meter is the strongly driven regime [62, 89]. The strong driving regime goes
beyond simple linear response theory of a driven qubit-cavity system, and in fact allows
testing of the Jaynes-Cummings spectrum. We will describe how the experiment provides a
remarkably excellent understanding of the Hamiltonian and circuit QED system in general.
The extraction of a limit on the cavity population is a nice extra result from this work. This
experiment is termed the nonlinear vacuum Rabi, and it will be described in detail here.

6.1.1 Nonlinear vacuum Rabi

In standard cavity QED experiments, the signature feature of the Jaynes-Cummings interac-
tion,
1 ħω a
H = ħωC (a † a + ) + σz + ħ (a† σ− + aσ+ ) (6.1)
2 2
is the vacuum Rabi splitting, as shown in figure 6.1. Vacuum Rabi peaks refer to the split
structure of the cavity transmission line when the atom is resonant with the cavity. The degree
of splitting is twice the coupling strength, 2, and is often termed the vacuum Rabi frequency.
Experimental observation of the splitting is dependent on the separation between the peaks
being much greater than the linewidth of the peaks.
The Jaynes Cummings interaction is not simply a single excitation construct, but in fact
describes an entire ladder of eigenstates, as shown in figure 6.2. By compounding a number of
√
resonant excitation pulses, it is possible to climb up the ladder and observe the 2 n spacing
√
between the levels, as previously discussed in section 3.3.1. The n splittings have been
observed in other circuit QED experiments as well [111], even allowing for the generation of
Fock states and other non-classical states of light [112, 113].
134 initialization and benchmarking

1.0
0.8
∣T∣ / ∣T ∣ /π = . MHz
0.6
0.4
0.2
0.0
6.86 6.88 6.90 6.92 6.94 6.96
Frequency (GHz)
Figure 6.1: Vacuum Rabi splitting. When the qubit is in strong coupling with the cavity, the
cavity transmission undergoes a splitting into two peaks, separated in frequency by /π. Shown
here is the normalized transmitted homodyne amplitude as a function of applied frequency,
with the vacuum Rabi peaks split by 94.4 MHz. This splitting is associated with one of the
qubits in sample cQED187.
photon number

energy

qubit level

Figure 6.2: Jaynes-Cummings ladder. The energy levels of the qubit-cavity coupled system
can be shown in this ladder diagram. On the left in blue are the energy levels corresponding to
the coupling turned off, with two ladders of increasing photon number with the qubit in either
state, ∣n, 0⟩ and ∣n, 1⟩. The red ladder of energy levels on the right reflect the strong coupling
√
interaction , resulting in a splitting of the resonant doublets ∣n, 0⟩ and ∣n − 1, 1⟩ by 2 n.
 6.1. initializing pure states 135

However, in the nonlinear vacuum Rabi experiment, we are able to investigate the Jaynes
Cummings ladder with a different approach. Instead of multiple time-synchronized pulses
which allow one to climb up to any level while starting in the ground state with no excitations
∣n = 0⟩, we use a continuous strong driving technique which performs an n-excitation virtual
transition up to the nth Jaynes-Cummings state.

Experimental procedure

The sample used for this experiment is cQED187. The fabrication and details of the sample
are previously discussed in chapter 5. Here, we simply re-state a few of the salient charac-
teristics necessary for understanding the strong driving experiment. The cavity has a λ/2
resonant frequency of 6.92 GHz with a photon decay rate of κ/2π = 300 kHz. Although the
sample contains two qubits, we will only be studying the balanced transmon (section 5.3.2),
corresponding to /π = 94.4 MHz and charging energy EC /2π = 340 MHz.
Time domain measurements of the balanced transmon show that T1 is limited by the
multimode Purcell effect and completely homogeneously broadened (T2 = 2T1 ) at the flux
sweet spot, where the maximal transition frequency is fmax = 7.48 GHz. The experiment
investigates the vacuum Rabi splitting, where the qubit is tuned into resonance with the
cavity. When the qubit is tuned slightly below the cavity frequency, at around 6 GHz, the
measured coherence times are T1 = 1.7 µs and T2 = 0.7 µs.
The experiment is performed via a heterodyne detection scheme [53]. An RF drive
tone is applied into the input side of the cavity. The frequency and power of this RF drive
are controllable at room temperature. The transmitted RF voltage signal from the cavity is
amplified both cryogenically and at room temperature before being mixed down to a 1 MHz
IF signal. The in-phase and quadrature components of the IF signal are extracted digitally.
These components are then combined as a heterodyne amplitude.
To accurately find the vacuum Rabi splitting, large transmission maps are taken as a
function of changing the applied transmission frequency and varying the external magnetic
field. Figure 6.3 gives a coarse location of splittings in the full two-qubit system, and provides
a way of finding the range of magnetic fields in which the vacuum Rabi splitting occurs.

Strong driving vacuum Rabi spectrum

When the qubit and cavity are exactly resonant, the cavity transmission peak is split into
two peaks with equal maximum transmitted homodyne amplitude, as shown in figure 6.1.
136 initialization and benchmarking

7.2
Qubit qR Qubit qL
crossing crossing
ωd /π [GHz]

7.0

6.8

6.6

6.4
-15 -10 -5 0 5 10 15
Magnetic Field B [a.u.]

Figure 6.3: Transmission versus magnetic field and drive frequency. Although the sample
cQED187 is a two transmon device, the strong driving vacuum Rabi experiment is performed
on only one of the transmons, specifically investigating the splitting shown in the red box,
around magnetic field B = 15.

However, upon turning up the power of the applied drive, the measured spectrum changes
into a more complicated structure. Figure 6.4 shows the emergence of other peaks in addition
to the original two in the vacuum Rabi splitting. Refs. [62, 89] describe this strong driving
effect and the supersplitting of the vacuum Rabi peak in greater detail.
Figure 6.4 also shows three slices through the power map, with each dip corresponding to
higher order photon transitions up the Jaynes-Cummings ladder. Solid lines in figure 6.4c-d
represent theory lines going directly through the experimental data. We will use these theory
fits to understand the full Jaynes-Cummings spectrum as well as infer a temperature of the
system.

Modeling the bath and multi-dimensional fit

To understand the measurement, we use input-output theory. Let the output bath mode be
described by the annihilation operator bout [97]. This bath mode can then be related to the
photons inside of the cavity by
√
bout = κa, (6.2)
 6.1. initializing pure states 137

a ∣+ ⟩
√
 
∣− ⟩
∣+ ⟩
√
∣− ⟩
 
∣+ ⟩
√
∣− ⟩
 

Energy
∣+ ⟩
√
∣− ⟩  
√
∣+ ⟩  
∣− ⟩
∣+ ⟩
∣− ⟩


Drive power [dB] ∣⟩

-10

-20
b
-30
0.2
c  dB
0.1
Transmitted Intensity A [A ]

0.2
d  dB
0.1

0.2
e − dB
0.1

6.85 6.90 6.95 7.00

ωd /π [GHz]
√
Figure 6.4: Emergence of n peaks under strong driving of the vacuum Rabi transition.
(a) The extended Jaynes–Cummings energy spectrum. All levels are shown to√scale in the
left part of the diagram: black lines represent levels ∣n, ±⟩ ≃ (∣n, 0⟩ ± ∣n − 1, 1⟩)/ 2 with only
small contributions from higher ( j > 1) transmon states; grey lines represent levels with large
√
contributions from higher transmon states. In the right part of the diagram, the n scaling of
the splitting between the ∣n, ±⟩ states is exaggerated for clarity, and the transitions observed in
plots (b–e) are indicated at the x-coordinate E n± /2πn of their n-photon transition frequency
from the ground state. (b) Measured intensity (A2 , heterodyne amplitude squared) in color
scale as a function of drive frequency and power. The multiphoton transitions shown in (a)
are observed at their calculated positions. (c–e), Examples of cuts for constant power, at the
values indicated in (b) (results from a master equation in black; experimental results in red),
demonstrating excellent agreement between theory and experiment, which is reinforced in
the enlarged insets. Good agreement is found over the full range in drive power from −45 dB
to +3 dB, for a single set of parameters. Figure reproduced from [62].
138 initialization and benchmarking

where we have dropped the reflected wave bin because the quantum noise contributes negli-
gibly to the classical noise due to the HEMT amplifier [62]. The measured output voltage
can be written in terms of the bath operators as VH = ⟨bout + bout †
⟩. After being amplified
with gain α, the voltage wave is mixed with a local oscillator of frequency ω LO , resulting in a
mixer output of

Vm = α⟨bout + bout ⟩† cos ω LO t (6.3a)

√
= α κ⟨ae −iωd t + a † e iωd t ⟩ cos ω LO t (6.3b)
√
α κ
= . (6.3c)
2
The final output is given at the intermediate frequency ω IF = ωd − ω LO , which in the experi-
ments we choose to be 1 MHz. Low pass filtering removes the fast oscillating terms, and then
we can extract the quadrature measurements

I = V0 ⟨a + a† ⟩
(6.4)
Q = V0 ⟨ia† − ia⟩,

where V0 is the voltage related to the gain of the entire experimental amplification chain. In
these experiments, the phase relation between the LO and the RF drive is not maintained while
sweeping the drive frequency. As a result, our detection scheme deals with the transmission
amplitude, given by
√
A= I 2 + Q 2 = 2V0 ∣⟨a⟩∣ = 2V0 ∣ tr(aρs )∣, (6.5)

where ρs is the steady-state density matrix of the system.

The actual experiment can be better modeled by using

A2 = ∣2V0 tr(aρs ) + bξ∣2 + 2σn2 , (6.6)

with b ∈ C describing the ampitude and phase of a direct leakage channel for the drive
to bypass the cavity, and σn is measurement noise in each of the I and Q channels. The
steady-state density matrix ρs is obtained from solving the master equation of the system
numerically. Details of the full master equation will not be given here and can be found in
[62]. Although the set of fit parameters is large, including V0 , b, EC , EJmax , ωC , T, Φ̃, ωd , ξ, κ,
γ1 , γ ϕ , most of the fit parameters can be measured to some degree in separate experiments,
and only slight adjustments are necessary here. The fits are obtained by minimizing the mean
 6.1. initializing pure states 139

squared deviation between the experiment and the model over the entire power range and
frequency range, with the only unconstrained fit parameters being b and two scaling factors
which describe the attenuation and amplification on the input and output signals. These fits
are shown in the solid-lines of the slices in figure 6.4.

6.1.2 Cavity temperature

The system temperature is related to the fitted values of κ+ and γ+ , both of which are consistent
with zero. The key figure of merit is the ratio r = κ+ /κ− = γ+ /γ− . In the data from figure 6.4,
the largest value of r placed into the fits which is still consistent is approximately 0.003,
associated with an upper bound on the reservoir temperature of ∼ 55 mK. This is higher than
the base temperature of the fridge, but it is still the most stringent bound on the temperature
of any circuit QED sample performed thus far.
To recognize the sensitivity to temperature which this technique provides, we can in-
vestigate strong driving with an elevated temperature. The same experiment as is shown in
figure 6.4 is performed, but with the circulator on the output of the cavity thermalized to
the ‘100 mK stage’ of the fridge, instead of the base plate. The actual temperature of this plate
is often in the 90–110 mK range. Now, a representative measurement is shown in figure 6.5,
where the primary new feature is the pair of broad peaks in between all of the multiphoton
peaks. We find good agreement with the theory when using an effective temperature of
130 mK. The fits get worse for larger drive powers due to the truncation of the Hilbert space
used in the master equation simulations.
The extra thermal peaks are due to overlapping transitions between very highly excited
states. With a high enough temperature, the strong driving thus causes a bistable situation,
where in one case we probe the anharmonic Jaynes-Cummings ladder, and in another case
where the system fluctuates into a sufficiently excited state that the anharmonicity is reduced
causing the strong driving to generate states comparable to coherent states. We can think of
the second case as a cascade effect, where strongly driving a small starting thermal population
can result in a highly excited state of the cavity. This effect makes this strongly driven vacuum
Rabi technique a particularly sensitive meter for the temperature of the cavity.

6.1.3 Vacuum Rabi summary

From the strong driving vacuum Rabi experiments, we can confidently say that our circuit
QED system operates with ⟨n⟩ ≲ 0.003. This implies that the qubits, which are detuned
140 initialization and benchmarking

a  dB
3

Transmitted Amplitude A [a.u.] 1

b − dB
1

0
1
c − dB

0
6.80 6.85 6.90 6.95 7.00 7.05
ωd /π [GHz]

Figure 6.5: Strongly-driven vacuum Rabi response at elevated temperature. For this run of
the experiment, the 50 Ω termination on the circulator at the output port of the sample was kept
at a temperature of ∼ 110 mK. The theoretical response (black) was calculated for an effective
temperature of 130 mK, showing good agreement with moderate driving, (c). For the stronger
driving of (a) and (b), the theory and experiment disagree due to the truncated Hilbert space
used in the simulations. Figure reproduced from [62].

from the cavity, should effectively start from a pure ground state ∣0⟩, so long as the qubits
are Purcell limited and thus coupled to the “cold” cavity bath. This is the case because the
temperature, which goes as γ+ /γ− does not change with the qubits detuned from the qubit.
Although both the rates γ+ and γ− decrease as 2 /∆2 (assuming a naive two-level qubit case),
the overall ratio, and hence the temperature stay the same.
Of course, there are situations where the qubit might be more strongly coupled to some
other loss mechanism (such as the intrinsic Q ∼ 50, 000 − 70, 000 discussed previously in
section 3.5.1) than the microwave resonator, in which case we would not be able to definitively
comment on the qubit starting state.
Nonetheless, in the multi-mode Purcell limit, we have then an upper bound on the
 6.2. characterizing single-qubit gates 141

number of photons in the cavity of 0.003 giving also an upper bound on the excited state
population P1 ∼ 0.003. This corresponds thus to a steady-state ground state population of
P0 = 0.997.
The quality of our ground states will be investigated in a different way later in this thesis,
when we perform state tomography on two qubit states in chapter 8. However, next in this
chapter, we will work with a single-qubit again, and benchmark the single-qubit gate fidelity.

6.2 Characterizing single-qubit gates

Gate fidelity, previously defined in (4.22), is the standard measure of agreement between
an ideal operation and its experimental realization. Beyond the gate fidelity, identifying the
nature of the dominant errors in a specific architecture is particularly important for improving
performance. While NMR, linear optics, and trapped ion systems are primarily limited by
systematic errors such as spatial inhomogeneities and imperfect calibration [35, 114, 115],
for solid-state systems decoherence is generally the limiting factor. It is thus crucial to
employ experimental tests of qubit operations which either distinguish between various error
mechanisms, or average over all the errors such as not to give a biased result. This question
of how to measure average gate errors or distinguish between various error mechanisms has
produced different experimental protocols for measuring gate fidelity, such as the double
π metric employed in superconducting qubits [116], process tomography as demonstrated
in trapped ions, NMR, and superconducting systems [35, 114, 115, 117], and randomized
benchmarking, as performed in trapped ions and NMR [118, 119].
The double π metric (π − π) is one of the simplest gate fidelity metrics, as it consists
of applying only two π pulses in succession. This should ideally correspond to the identity
operation 1. The aim of π-π is to determine the deviations from 1 by measuring the residual
population of the excited state following the pulses. Despite its simplicity, this metric captures
the effects of qubit relaxation and the existence of levels beyond a two-level Hilbert space.
However, in general, it is merely a rough estimate of the actual gate fidelity as it does not
contain information about all possible errors. In particular, errors that affect only eigenstates
of σx or σ y and deviations of the rotation angle from π are not well captured by this measure.
A second metric that, in principle, completely reveals the nature of all deviations from
the ideal gate operation is Quantum Process Tomography (QPT) [120]. Ideally, QPT makes it
possible to associate deviations with specific error sources, such as decoherence effects or
non-ideal gate pulse calibration. However, in systems where the “preparation,” “process,” and
 142 initialization and benchmarking

“measurement” all involve the same single-qubit rotations, it is difficult to assign the results
from QPT to a single gate error. Moreover, the number of measurements that are necessary
for QPT scales exponentially with the number of qubits.
While QPT provides information about a single gate, randomized benchmarking (RB)
[118, 121] gives a measure of the accumulated error over a long sequence of gates. This
metric hypothesizes that with a sequence of randomly chosen Clifford group generators
(Ru = e ±iσu π/4 , u = x, y) the noise can behave as a depolarizing channel where all error
mechanisms are weighted equally, and an average gate fidelity can be obtained. In contrast to
both π − π and QPT, RB is approximately independent of errors in the state preparation and
measurement. Also, while the other metrics measure a single operation and extrapolate the
performance of a real quantum computation, RB tests the concatenation of many operations
(here up to ∼ 200), just as would be required in a real quantum algorithm.
In the rest of the chapter, we present measurements of single-qubit gate fidelities where the
three metrics mentioned above are implemented in our circuit QED system with a transmon
qubit. We find single-qubit gate errors at the 1 ∼ 2% level consistently among all metrics.
These low gate errors reflect the good coherence times [63, 82], systematic microwave pulse
calibration, and accurate determination of gate errors despite limited measurement fidelity.
Specifically, in circuit QED, measurement fidelity can be as high as 70%, though in this
experiment it is ∼ 5%, as readout is not optimized. Although the experiments are performed
in a solid-state qubit implementation, the theory and discussion about the gate errors in this
chapter extend generally to all qubit systems including ions and spins. Before going into
details on each of the metrics, let us first describe some basic experimental information about
the sample and pulse calibration.

6.3 Single-qubit gate error experiments

The gate error protocols are performed on cQED187, which is described in detail in chapter 5.
Although the sample consists of two transmon qubits coupled to a coplanar waveguide
resonator, we investigate the gate fidelity of only the balanced transmon, with the other
unbalanced transmon tuned away from any interaction. Experimentally measured parameters
include the qubit-cavity coupling strength given by 0 /π = 94.4 MHz, the resonator frequency
ωr /2π = 6.92 GHz, photon decay rate of κ/2π = 300 kHz, and qubit charging energy EC /2π =
340 MHz. The qubit is detuned from its flux sweet spot by ∼ 1.5 GHz with a resonant frequency
of ω01 /2π = 5.96 GHz, and coherence times of T1 = 2.2 µs and T2∗ = 1.3 µs.
 6.3. single-qubit gate error experiments 143

(a) σ
clock cycle

tgate

tbuffer

σ

(b)
0.1
Amplitude (V)

0.0

-0.1

0 10 20 30 40
Time (ns)
Figure 6.6: Microwave pulse shapes. The applied pulse shapes (a) are Gaussians, with standard
deviation given by σ, and truncated on each side to take up a total time of 4σ. In the experiments,
σ is typically between 1–12 ns. A delay of tbu f f er , typically 5–8 ns is included at the end of
each truncated pulse shape to allow for complete turn off, as the generated pulse shapes in the
AWG result in a spurious tail on the falling edge. One clock cycle corresponds to the total gate
time t  . (b) Measured pulses with σ = 3 ns on a fast-sampling scope after modulation with a
microwave frequency signal at 5 GHz. The residual incomplete pulse turn-off can be seen after
the falling edge of each pulse.

6.3.1 Microwave pulse shaping

As previously discussed in section 4.2, microwave frequency with quadrature control can
be used to perform rotations about the x and y axes of a single-qubit. Rotations about any
axis of the Bloch sphere can be generated from combinations of rotations around x and
y. The carrier frequency of the microwaves need to be resonant with the qubit transition
frequency and the pulse amplitudes and phases define the rotation angle and axis orientation,
respectively. In all experiments, the pulse-shape is Gaussian with standard deviation σ that
we vary between 1 and 12 ns. The pulses are truncated at 2σ on each side and a constant buffer
time of 8 ns is inserted after each pulse to ensure complete separation of the pulses. The
Gaussian pulse envelopes are generated with a 10-bit Tektronix AWG520 arbitrary waveform
144 initialization and benchmarking

0.10

0.00
V

-0.10

0 100 200 300 400

Time [ns]
Figure 6.7: Sample sequence of concatenated pulses. Example of a sequence of concatenated
pulses for randomized benchmarking section 6.3.6. Blue (red) pulses are x (y) gates of either π
or π/2 rotations. The longest applied sequences consist of ∼ 200 gates.

generator with 1 ns resolution. The pulse shapes are mixed with sine and cosine waves at the
qubit transition frequency of 5.95 GHz using an Agilent E8267C Vector Signal generator. It
is thus possible to produce pulses phase-shifted by 90○ for qubit rotations around x and y.
Figure 6.6 shows two sample pulses which we program into the AWG and the microwave
modulated single-qubit pulses used in our experiments.
When experimentally observing each pulse with a fast oscilloscope, it becomes clear
that each pulse turns on much quicker than it turns off, with a residual tail that which
takes up around ∼ 4 − 8 ns. The inclusion of the relatively long buffer time of 8 ns at
the end of each pulse is to ensure that a single microwave pulse is completely turned
off before the next one is applied. The gate characterization experiments can involve se-
quences of up to ∼ 200 concatenated pulses, such that avoiding residual pulse overlap be-
comes very important. Figure 6.7 is a sample sequence showing a train of concatenated
pulses measured on the fast-sampling scope. The gate sequence corresponds to applying
R y (π)R y (−π/2)Rz (−π)R x (π/2)... R y (−π/2)R x (π)Rz (π)R y (−π).
Rotations about the z axis are performed with a rotation of the reference frame with
an accompanying delay equivalent to the time required for x and y pulses, see Ref. [118].
For example, the sequence R x (π)Rz (π)R y (π) becomes R x (π)1R y (−π). Although this is
permissible for single-qubit experiments, for multiple qubits, a simple rotation of the frame is
not enough, and explicit z-operations can be performed by modulating each qubit’s transition
frequency, either by ac-Stark shift or flux bias.
 6.3. single-qubit gate error experiments 145

6.3.2 Calibration of single-qubit gates

In order to obtain the best gate fidelity, accurate pulse amplitude calibration is necessary.
We use tune-up sequences similar to those used in NMR experiments [122], which involve
applying repeated pulses such that small systematic errors accumulate to produce a large
signal which may then be nulled out. The rotations R x (±π/2), R y (±π/2), R x (±π), R y (±π)
are each calibrated independently.
To calibrate the π/2 rotation around the x axis, we apply sequences consisting of an odd
number n of the pulses, [R x (π/2)]n , and subsequently measure ⟨σz ⟩. Performing this for
several n, we calibrate the pulse amplitude by comparing the measured values with the ideal
outcomes, which generally should look identical, half way between the signal of performing
1 and R x (π). A slight over-rotation is the π/2 rotations results in a zig-zag looking pattern,
and the calibration experiment is tuned until this pattern becomes flattened. The other π/2
pulses are calibrated similarly.
Once all π/2 pulses are calibrated, the π pulses are calibrated with a slightly different
scheme. To calibrate a π pulse around the x-axis, we first apply a single calibrated R x (π/2)
pulse followed by an integer number m of the π pulses, R x (π/2)[R x (π)]m . Again, the mea-
sured values of ⟨σz ⟩ for different m are compared with simulation to calibrate the amplitude
of the R x (π) pulse. The prepended π/2 pulse has the effect of rendering the scheme to be
first-order sensitive to deviations in the π pulse amplitude. The scheme is repeated mutatis
mutandis for the other π pulses.

6.3.3 Single qubit readout calibration

The measurements performed for gate characterization are performed in the time-domain
and employ the strong dispersive regime of circuit QED. As discussed in section 3.4.2, in this
regime, we can apply microwaves at the frequency corresponding to the qubit in the ground
state to obtain a projective measurement on ∣0⟩. Here, we apply a square-pulse modulated
interrogation tone at ωC + χ, and detect both an in-phase and quadrature homodyne voltage
transient. In all the time-domain experiments, all qubit operations are applied before the
measurement pulse. The same gate sequence is applied repeatedly roughly 250,000 times, and
the recorded measurement transients are averaged to combat the low ∼ 5% readout fidelity.
We then integrate the measured transient signal over a time t ∼ T1 after the measurement
tone is turned on to obtain an average homodyne voltage response in both quadratures. These
two quadrature voltages, I and Q, are combined with a quadrature sum to form an average
146 initialization and benchmarking

√
homodyne voltage amplitude AH (ρ) = I 2 + Q 2 , which is a function of the single-qubit state
ρ.
We calibrate the single qubit measurement by finding AH corresponding to having pre-
pared the states ∣0⟩ and ∣1⟩. State ∣0⟩ is simple as it involves applying no microwaves to excite
the qubit. A coarse calibration of the qubit pulse amplitude needed to prepare ∣1⟩ is found by
performing a Rabi oscillation in amplitude: apply a varying amplitude of a Gaussian shaped
pulse at the qubit frequency and measure AH for each amplitude. The maximum deviation of
A H from A H (∣0⟩ ⟨0∣) is nominally the amplitude for ∣1⟩. Then, finer tuning is then performed
through implementing the single-qubit rotation calibration sequences mentioned in the
previous section. Therefore, the accurate single-qubit readout calibration and single-qubit
gate calibration bootstrap off one another.
With the level of AH defined for both ∣0⟩ and ∣1⟩, the population of the excited state P1 for
any prepared single-qubit state ∣ψ⟩ is determined by a simple normalization,

AH (ρ) − AH (∣0⟩ ⟨0∣)

P1 = . (6.7)
AH (∣1⟩ ⟨1∣) − AH (∣0⟩ ⟨0∣)

In all experiments, we concatenate readout calibration experiments which involve either

applying no microwaves (to prepare ∣0⟩) or R x,y (π) (to prepare ∣1⟩) and extract P1 . These
readout calibration experiments are necessary as a result of phase drift on the homodyne
detection, which can result in the overall amplitude of the homodyne voltage to change over
time. The measurements in the rest of the chapter will correspond to determining either P1
or P0 = 1 − P1 .

6.3.4 Double π metric

After calibrating our single-qubit pulses, we perform the π-π experiments by applying a π
pulse, waiting for a separation time tsep , applying a second π pulse, and then performing
the homodyne measurement described from the previous section, to find the excited state
probability P1 . Figure 6.8a shows the measured P1 for the π-π experiment as a function of
tsep using σ=2 ns pulses. Due to the decay of the excited state following the first π pulse, P1
increases as a function of tsep .
The effect of the experiment can be accurately captured in simulations with a simple
theoretical model consisting of the dynamics from a master equation for a driven three-level
atom subject to relaxation and dephasing, with corresponding time-scales T1 and Tϕ . The
 6.3. single-qubit gate error experiments 147

(a) 0.25
0 5 10 15 20 25

0.02
0.20
0.00

0.15
P1

0.10

0.05

0.00
0 100 200 300 400
tsep [ns]
(b) 1.0
0.8
0.6
P1

0.4
0.2
0.0
0 20 40 60 80
Pulse Length [ns]

Figure 6.8: Bang-bang gate characterization and visibility. (a) Excited state qubit population
P1 vs. separation time tsep between two successive π-pulses (σ = 2 ns). The data agree well
with the simulation (solid line) involving relaxation and decoherence. The inset shows addi-
tional data taken for 0 ≤ tsep ≤ 30 ns. The residual population corresponding to the minimal
separation is found to be 0.014 ± 0.008 giving a single qubit gate error of 0.7 ± 0.4%. (b) Rabi
oscillations show a visibility of 100.4 ± 1.0%.
148 initialization and benchmarking

coherent evolution is governed by the Hamiltonian

H = ħ ∑ [ω0 j σ j† σ j + ε j (t)(σ j† + σ j )] , (6.8)

j=1,2

where σ j = ∣ j − 1⟩ ⟨ j∣ is the lowering operator for the multi-level atom with eigenenergies ħω j .
The corresponding transition energies are denoted ħω i j = ħ(ω j − ω i ). Drive strength and
pulse-shapes are determined by

2j
ε j (t) = [X(t) cos(ωd t) + Y(t) sin(ωd t)] . (6.9)
ωr − ω j−1, j
√
Here,  j ∼ j0 is the transmon coupling strength [61], ωd /2π is the frequency of the drive,
and X(t) and Y(t) are the pulse envelopes in the two quadratures.
The inset of figure 6.8a shows the experiment with tsep varying between 0 ns and 30 ns
repeated 2.5×106 times. We measure P1 = 0.014±0.008 at tsep = 0 ns. Dividing this probability
by two as in Ref. [116] gives a single gate error of 0.7 ± 0.4%.
Conceptually, the π-π measure is similar to the visibility measure used by Wallraff et
al. in Ref. [123], corresponding to (1 − ⟨σz ⟩)/2 after a single π pulse. Figure 6.8b shows Rabi
oscillations made by increasing the length of a pulse resonant with the qubit transition
frequency. The visibility is found to be 100.4 ± 1.0%. This also agrees with our simple
theoretical model taking into account the T1 , T2∗ , and third-level at our specific operating
point.
Although the π-π measure is relatively simple to implement, it does not sufficiently
take into account errors which may manifest when the qubit state is neither ∣0⟩ nor ∣1⟩. For
√
example, qubit dephasing significantly influences superposition states such as (∣0⟩ + ∣1⟩)/ 2.
However, the π-π scheme does not involve any π/2 rotations which would be required to
generate such states. Furthermore, although we calibrate our π and π/2 pulses, deviations of
the rotation angle only manifest as second-order errors in π pulses, as opposed to linearly
in π/2 pulses. Although, there are experiments which refer to this measurement of 1 − P1 as
a gate fidelity [116], in practice, due to the incompleteness of the protocol, we simply take
the gate error result of π-π as a simple estimator, with more stringent tests necessary to fully
characterize the single-qubit operations.
 6.3. single-qubit gate error experiments 149

Initialize Apply Process State Tomography

Homodyne
measurement

1 1 1
R x (π/) R x (π/) R x (π/)
R y (π/) R y (π/) R y (π/)
R y (π) R y (π)

Figure 6.9: Schematic for quantum process tomography. QPT consists of three stages of gates.
The initialization
√ stage
√ involves rotation gates which prepare the input states ∣0⟩ , ∣1⟩ (∣0⟩ −
∣1⟩)/ 2 (∣0⟩ + i ∣1⟩)/ 2. The second stage involve applying the process to be studied, which in
the experiments presented here are 1, R x (π/2), and R y (π/2). The final stage performs state
tomography of the qubit system by measuring the projection along the three Cartesian axes
through the application of 1, R x (π/2), R y (π/2), and R y (π). This is followed by the homodyne
measurement described in section 6.3.3.

6.3.5 Quantum process tomography

The idea behind QPT is to determine the completely positive map E, which represents the
process acting on an arbitrary input state ρ. The theory is detailed in Refs. [120, 124] and can
be summarized as follows. Any process for a d dimensional system can be written as
d 2 −1
E(ρ) = ∑ χ mn B m ρB†n , (6.10)
m,n=0

where {B n } are operators which form a basis in the space of d ×d matrices, and χ is the process
matrix that we aim to measure. Here, any d × d matrix can be written as linear combinations
of the elements of {B n }. The process matrix χ is a positive superoperator (a linear map
of a space of operators to another space of operators) which completely characterizes the
process E using the basis operators {B n }. To determine χ, we prepare d 2 linearly independent
input states {ρinn }. For every input state, the output state ρ n = E(ρ n ) is determined by
out in

state tomography (section 2.5.2). The process matrix is then obtained by inverting Eq. (6.10),
although in general this last step does not guarantee a completely positive map. To remedy
this, a maximum likelihood estimation (MLE) based on Ref. [35] can be used.
In all the QPT experiments, the measurements are performed after sequences of three
concatenated pulses (figure 6.9) are applied to the qubit. The first pulse, chosen from {1,
150 initialization and benchmarking

R x (π), R x (π/2), R y (π/2)}, prepares the four linearly independent input states ∣0⟩ , ∣1⟩ , (∣0⟩ +
√ √
i ∣1⟩)/ 2, and (∣0⟩ − ∣1⟩)/ 2, whose projectors span the space of 2 × 2 density matrices
ρ. The second pulse corresponds to the process for which look to determine χ, and is
chosen from, {1, R x (π/2), R y (π/2)}. A final pulse ({1, R x (π), R x (π/2), R y (π/2)}) rotates
the measurement axis to perform state tomography on the state resulting from the first two
pulses.
The state tomography data allows us to construct the process E(ρ), for one qubit (d = 2),
and find the process matrix χ mn which is defined with respect to the operator basis given
by the Pauli basis {B n } = {1, σx , σ y , σz }. By definition the χ matrix must be Hermitian.
Furthermore, the completeness constraint requires that it must satisfy [120]

∑ χ mn B†n B m = 1. (6.11)
mn

To find χ using MLE, we first write the process matrix in a Cholesky decomposition of the
form

χ(⃗t ) = T † T, (6.12)

where T is a lower triangular matrix parametrized by the vector ⃗t . This ensures that χ be
Hermitian. Next, the measured data is fit to a physical process by minimizing the function

d2 ⎡ d 2 −1 ⎤2
⃗ ⎢ † ⎥
f (t ) = ∑ ⎢m ab − ∑ χ mn Tr[Mb B m ∣ϕ a ⟩ ⟨ϕ a ∣ B n ]⎥ . (6.13)
⎢
a,b=1 ⎣
⎥
m,n=0 ⎦
Here, m ab is the measured data for the case where the state ∣ϕ a ⟩ was prepared and the
observable Mb was measured. A Lagrange multiplier is then used to impose the completeness
condition (6.11)], such that we find the minimum of the function

d2 ⎡ d 2 −1 ⎤2
⎢ ⎥
f (⃗t ) = ∑ ⎢m ab − ∑ χ mn Tr[Mb B m ∣ϕ a ⟩ ⟨ϕ a ∣ B n ]⎥
†
⎢
a,b=1 ⎣
⎥
m,n=0 ⎦
(6.14)
d −1 ⎡ d −1 ⎤
2 2 2
⎢ ⎥
+ λ ∑ ⎢ ∑ χ mn Tr[B m B k B†n ] − Tr[B k ]⎥
⎢
k=0 ⎣m,n=0
⎥
⎦
to obtain the most probable completely positive χ matrix corresponding to the measured
values.
The results of QPT on the three processes 1, R x (π/2) and R y (π/2) are shown in figure 6.10.
 6.3. single-qubit gate error experiments 151

Re[χ] Im[χ]
(a)

1 1.0 0.5

0.5 0.0

σz σz
0.0 σy σy
- 0.5
1 σx 1 σx
σx σx
σy σy 1
σz 1 σz

(b) 0.5
1.0

R x (π/)
0.5 0.0

σz σz
0.0 σy -0.5 σy
1 σx 1 σx
σx σx
σy 1 σy 1
σz σz

1.0 0.5
(c)
R y (π/) 0.5 0.0

σz σz
0.0 σy -0.5 σy
1 σx 1 σx
σx σx
σy 1 σy
σz σz 1

Figure 6.10: Quantum process tomography experimental results. Real and imaginary parts
of the experimentally obtained process matrix χ for the three processes (a) 1, (b) R x (π/2), and
(c) R y (π/2) for σ = 2 ns.
152 initialization and benchmarking

0.08 1
( − F  ) R x (π/)
0.06 R y (π/)

0.04
Gate Error

0.02

0.00 theory for R x (π/)

10 20 30 40 50 60
Total Pulse Length [ns]
Figure 6.11: Gate errors from QPT. Gate error vs. total pulse length obtained from quantum
process tomography plotted for the three processes 1, R x (π/2), R y (π/2)

Here, bar plots of the real and imaginary parts of χ are shown for a pulse with σ = 2 ns in the
Pauli basis {B n } = {1, σx , σ y , σz }. We can compare our data to the ideal process matrices χideal .
For instance, for the 1 process, we expect χ11 = 1 and χuu′ = 0 otherwise, which is in good
agreement with the measured results. Small deviations from χideal arise from preparation and
measurement errors, gate over-rotations, decoherence processes, qubit anharmonicity, etc.
Calibration errors of the rotations around the x axis are seen as a non-zero Im[χ1σx ] and a
drive detuning error is exhibited in Im[χ1σz ].
From the experimentally obtained process matrix χ and its ideal counterpart χideal we
can directly calculate the process fidelity, defined as F p = tr[χideal χ], as well as the gate fidelity
F  = ∫ dψ ⟨ψ∣ U † E(ψ)U ∣ψ⟩. Here the integral uses the uniform measure dψ on the state
space, normalized such that ∫ dψ = 1. F  can be understood as how close E comes to the
implementation of the unitary U when averaged over all possible input states ∣ψ⟩. From
Ref. [125], there is a simple relationship between the F p and F  , namely F  = (dF p + 1)/(1 + d).
For the three processes displayed in figure 6.10, F p is 0.96, 0.95, and 0.95 ±0.01.
Figure 6.11 shows F  as a function of pulse length. The error bars are standard deviations
obtained by repeating the maximum-likelihood estimation for input values chosen from a
distribution with mean and variance given by measurement. The large scatter in the gate
error versus pulse length is primarily attributed to drift in the system which occurs over the
course of the data acquisition. Such errors can be reduced via cryogenic magnetic shielding,
but was not done for the experiment presented here.
 6.3. single-qubit gate error experiments 153

Although QPT is an excellent way to establish the complete behavior of the quantum gate
for certain systems where preparation and readout are near perfect and independent of the
process operation, in circuit QED, it does not extricate the process errors from errors in the
preparation or analysis gates. Specifically, in the experiment described above, the demon-
strated gate fidelity corresponds to in each case the performance of a three gate sequence. As
a result, QPT as a gate error technique to some degree gives too much information which
is not simply attributable to any specific gate or syndrome. Furthermore, as the number
of qubits increases, the complexity of process tomography scales exponentially due to the
increased number of input states as well as basis states for the state tomography, making it a
less attractive option for error determination in larger quantum systems.
Instead of trying to find out the all the details of the errors in the system via QPT, a
different approach is to find an average gate fidelity. This can be done with randomized
benchmarking, which we detail in the next section.

6.3.6 Randomized benchmarking

The randomized benchmarking experimental protocol, described in Knill et al. [118], consists
of the following: (1) initialize the system in the ground state, (2) apply a sequence of randomly
chosen pulses in the pattern ∏i C i Pi where C i are Clifford group generators e ±iσu π/4 , with
u = x, y, and Pi are Pauli rotations, i.e. 1, σx , σ y , σz , (3) apply a final Clifford or Pauli pulse to
return to one of the eigenstates of σz , (4) perform repeated measurements of σz , and compare
to theory to obtain the sequence fidelity.
The RB sequences contain only π and π/2 rotations, which are in the Clifford group such
that the single qubit state is always an eigenstate of a Pauli operator during the pulse sequence.
The C i pulses are all π/2 pulses and the Pi pulses which follow are π pulses which serve to
randomize the errors. It also means that truncation of any long random sequence of such
gates places the qubit in a state which is at most a single π/2 rotation away from an eigenstate
of σz . The π pulses can be thought of as rotations which rotate the Cartesian axes of the Bloch
sphere into one another.
Randomization with Clifford gates has been shown [125] to provide a depolarized noise
channel. A depolarized channel is a type of quantum noise which takes the state of a qubit
towards a completely mixed state with a probability d. Specifically, the new state of a qubit in
154 initialization and benchmarking

state ρ after being acted on by a depolarized noise is given by [12]

1
ρ ′ = d (1 ) + (1 − d)ρ. (6.15)
2
The effect of this depolarized channel is to reduce the amplitude of the Bloch vector towards
the center of the Bloch sphere [12]. Qubit noise processes such as relaxation (T1 processes)
and decoherence (Tϕ processes) are not depolarizing, but polarizing, as it is easy to see
that relaxation tends to bring the state to ∣0⟩ and decoherence results in the state going
towards either ∣0⟩ or ∣1⟩. However, the randomized Clifford gates rotate the qubit Bloch vector
throughout the Bloch sphere, sampling different parts of the state space which can be more
or less sensitive to specific forms of noise. This accumulation of different types of errors will
end up giving an average effect of all the errors, and result in a depolarized noise channel.
Then, the fidelity of the system will on average decay exponentially with the number of gates
applied.
To see the exponential decay, we will repeat the derivation as given in Ref. [118]. Suppose
we have a long randomized sequence of L combinations of randomized Pi C i and perform
experiments at truncated lengths i = 1, ..., L. Let p i be the average error probability for se-
quences of length i and d i be the probability that at any given step i, the qubit has depolarized.
The probability of the state being depolarized at step k is given by
k
1
p k = E ( [1 − ∏(1 − d i )]) , (6.16)
2 i=0

where the expectation is taken over the distribution of random choices of Pi C i and the factor
of 1/2 emerges due to depolarization giving the correct state regardless 1/2 of the time. Since
all the choices of Pi C i are independently chosen except for the final pulse, we have
1
pk = [1 − (1 − d f )(1 − d)k ] , (6.17)
2
where d is the average depolarization probability of a random choice of Pi C i and d f is the
depolarization probability of the final pulse. From this result, we see that p k decays to 1/2
exponentially as a function of the number of gates k and the decay constant gives d. Then,
we finally have that the average gate fidelity F  is related to d via [118]

d
F = 1 − (6.18)
2
 6.3. single-qubit gate error experiments 155

Measure
Initialize Apply random sequence
polarization
Homodyne
measurement

Figure 6.12: Schematic for randomized benchmarking. The randomized benchmarking

scheme begins with the qubit always in the ground state ∣0⟩. That is followed by a randomized
sequence of alternating Clifford and Pauli gates. A final gate is always applied to place the
final state in either ∣0⟩ or ∣1⟩ such that the final measurement will be on an eigenstate of σz . As
exponential decay of the final state fidelity is found as a function of the number of gates in the
central randomized sequence of Clifford and Pauli gates.

We follow the experimental recipe for the pulse sequences exactly as given in Ref. [118]. We
create computational sequences 192 pulses long, with 4 different randomizations of Clifford
gates C i , and 8 different randomizations of Pauli gates Pi for 32 total unique sequences. Each
sequence is then truncated to 17 different computation (where a computation consists of two
gates, Pi C i ) lengths, {2, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96}. This therefore
results in 544 different experiments, each applied for 250,000 repeated measurements, taking
a total time of about an hour.
The experimental results show the average fidelity is an exponentially decaying function
with respect to the number of gates. Figure 6.13 plots the fidelity as a function of the number
of computational gates for all randomized sequences with σ = 3 ns. An average error per
gate of 0.011 ± 0.003 is obtained by averaging over all the randomizations and fitting to the
exponential decay. The excellent fit to a single exponential indicates a constant error per gate,
consistent with uncorrelated random gate errors due to T1 , Tϕ , and no other mechanisms
significantly affecting repeated application of single-qubit gates.
The benchmarking protocol is repeated for different pulse widths σ, and the average error
per gate is extracted for each. Figure 6.14 shows the average fidelity versus number of gates
for a number of pulse widths. The extracted average error per gate can be plotted versus total
gate length, and compared to theory, as shown in figure 6.15. For large σ, experimental results
agree well with theory. In this regime, errors are dominated by relaxation and dephasing. For
small σ, the gate fidelity is limited by the finite anharmonicity and the resulting occupation
of the third level, although the effect is not very pronounced. We obtain error bars from
standard deviations in error per gate having generated fidelity values from distributions with
means and variance obtained from the experiment and theory. We observe an increase in
156 initialization and benchmarking

1.0

0.8

0.7
Average Fidelity

0.6
exponential fit
0.55 total avg fidelity
clifford 1
clifford 2
0.52 clifford 3
clifford 4
0.5
0 20 40 60 80 100
Number of Computational Gates

Figure 6.13: Randomized benchmarking for 3 ns pulse width. Average fidelity vs. number of
applied computational gates for all 32 unique sequences are shown in the small gray points.
Computational gates consist of a randomized Pauli with a randomized Clifford generator. The
colored points involve averaging over the 8 Pauli randomizations. The black squares are the
result of then averaging over the 4 Clifford randomizations. We can see that just the averaging
over the randomized Pauli gates already gives an exponential decay. The solid black line is an
exponential fit, from which we find an average gate error of 1.1%.

the standard error for σ = 1 ns. This increased variance in the experiments we attribute to the
onset of the finite anharmonicity. The optimal gate length is found to be 20 ns, as shown in
Fig. 4(b). The anharmonicity effect will be explored further in the next section on a different
sample where the coupling and hence leakage to the second excited state is much stronger.
The reduction of the error by a factor of ∼ 1/3 from QPT is likely due to the over-estimation
of errors in QPT where gate errors cannot be isolated from measurement and preparation
errors.

6.3.7 Summary of error metrics

We have thus systematically investigated gate errors in a circuit QED system by measuring gate
fidelity using the π − π metric, quantum process tomography, and randomized benchmarking.
Table I summarizes our results and displays consistently low gate errors across all metrics.
 6.3. single-qubit gate error experiments 157

error per gate = 0.019 ± 0.003

1.0
σ = 9 ns
0.8

0.6

0.4
error per gate = 0.015 ± 0.003
1.0
σ = 5 ns
0.8
Fidelity = 1-Perror

0.6

0.4
error per gate = 0.012 ± 0.005
1.0
σ = 2 ns
0.8

0.6

0.4
error per gate = 0.013 ± 0.009
1.0
σ = 1 ns
0.8

0.6

0.4
20 40 60 80 20 40 60 80
Number of Computational Gates

Figure 6.14: Trace by trace randomized benchmarking. The average error per gate is shown
in the left panels for σ = 9, 5 2 1 ns. The 32 unique trace-by-trace realizations are shown for the
same σ in the right panels. We can see the increase in the spread of all the traces at the shortest
pulse lengths, reflected in the increase in the error bars of the extracted error per gate.
158 initialization and benchmarking

Pulse Width σ [ns]

1 2 3 4 5 6 7 8 9 10 11 12
0.03
Average Error Per Gate

0.02

0.01
experiment
theor y

0.00
10 20 30 40 50 60
Total Gate Length [ns]

Figure 6.15: Error per gate versus pulse width. Average error per gate (experimental and
theoretical) at different pulse widths. The rise for σ < 2 ns corresponds to the onset of limitation
by the third level of the transmon. The increase in error per gate for σ > 2 ns is due to the
limitation by relaxation..

Table 6.1: Gate errors for the three metrics. The measure-
ments show consistently low gate errors of the order of 1 ∼ 2%.

Metric Measured error in %

π–π 0.7 ± 0.4
Process tomography: 1 2.4 ± 1.1
Process tomography: R x (π/2) 2.6 ± 0.8
Process tomography: R y (π/2) 2.2 ± 0.7
Randomized benchmarking 1.1 ± 0.3

From comparison with theory, we conclude that the observed magnitude of errors fully agrees
with the limitations imposed by qubit decoherence and finite anharmonicity. Specifically,
in the T1 limited case and for moderate gate lengths t  , we find that the gate error scales as
∼ t  /T1 . Once coherence times of superconducting qubits and pulse-shaping are improved,
the aforementioned metrics will be useful tools for characterizing gate fidelities as they
approach the fault-tolerant threshold.
 6.4. derivative-based pulse shaping 159

Randomized benchmarking will be a particularly attractive option for multi-qubit sys-

tems due to its favorable scaling properties as compared to QPT. There have already been
experiments in NMR which investigate randomization for two-qubit gates as well [119]. We
next extend the RB single-qubit gate error characterization for a qubit which experiences a
sharp increase in error for short pulse lengths.

6.4 Derivative-based pulse shaping

In the previous section, our experimental characterization of single-qubit gates revealed

that the gate fidelity is limited by the coherence times of the system and how fast the gates
can be performed. Certainly one limit to how fast we can perform qubit operations is the
hardware used. Our current arbitrary waveform generators allow for pulses with ns resolution
(section 5.7). However, as discussed in section 4.2.3, for shorter and shorter pulses, errors
can also arise due to the finite and small anharmonicity of the transmon qubits.
The sample discussed in the previous section does not show a very sharp increase in the
gate error at the shortest pulses because the transmon transition energy levels are located
below the cavity and the coupling strength 01 is relatively small, both contributing to making
the direct coupling to the ∣2⟩ state negligible. However, now we present similar single-qubit
gate experiments but on one of the qubits in cQED222 (chapter 5) in which this coupling is
not so small, and the effect of the transmon anharmonicity becomes evident in the measured
average gate errors.

6.4.1 Experimental details

Again, although cQED222 contains two qubits coupled to a coplanar waveguide cavity, here
we only investigate single-qubit gates on one of the transmon qubits. The other qubit is
tuned via its local flux bias line far away from both the cavity and the qubit which we will be
studying. The bare cavity resonance frequency is 6.902 GHz, and the investigated transmon
is tuned to a maximal frequency ωmax = 8.18 GHz, with corresponding coherence times of
T1 = 1.2 µs and T2 = 1.5 µs. The anharmonicity of the system is governed by the charging
energy, measured via spectroscopic means to be EC /2π = 317 MHz [126, 127].
160 initialization and benchmarking

1. 0
σ = 6 ns (a) σ = 3 ns (b)
0. 9
exponential fit
0. 8 total avg fidelity
clifford 1 clifford 2
0. 7 clifford 3 clifford 4
0. 6
Fidelity = 1-PError

0. 5
1. 0
σ = 2 ns (c) σ = 1 ns (d)
0. 9
0. 8
0. 7
0. 6
0. 5
20 40 60 80 20 40 60 80
Number of Computational Gates

Figure 6.16: Clifford averaged RB for cQED222 with standard pulse shaping. Total average
fidelity (black squares) and averaging just over the Pauli randomizations, (red, green, cyan, blue
hash markers), for σ = 6, , 3, 2, 1 ns. At σ = 3 ns, the error per gate already starts to increase,
and at σ = 1 ns, the scatter even in the Pauli averaged traces remains large. The solid black lines
are exponential fits to the total average fidelities.

6.4.2 Results with standard pulse shaping

We first repeat the randomized benchmarking experiment of section 6.3.6 using truncated
Gaussian pulses of varying σ. The average fidelity is found to be an exponentially decaying
function for all σ. Figure 6.16 show the 4 Clifford randomizations with the 17 different
truncated lengths for σ = {6, 3, 2, 1} ns. We are able to see improvement in the average fidelity
measurements as σ goes from 6 ns to 3 ns (figure 6.16a–b). However, it is also obvious that
for shorter pulses, σ = 3 − 1 ns, the extracted average error per gate gets progressively worse
(figure 6.16b–d).
This is made more obvious by extracting the average gate fidelity from exponential fits
for each of the different σ, plotted in figure 6.17. Furthermore, the calculated standard errors
(shown as error bars at each point) for each σ can also be seen to get worse with shorter gate
 6.4. derivative-based pulse shaping 161

Pulse Width σ (ns)

1 2 3 4 6

0. 1
6
4
Average Error Per Gate

0.01
6
4 RB with gaussians
2
simple T 1 theory

0.001
10 15 20 25
Total Gate Length (ns)
Figure 6.17: Error per gate with normal pulse shaping. Extracted error per gate versus total
gate length using Gaussian pulse shapes. There is a sharp rise in the error per gate at σ = 3 ns.
The solid line is a simple two-level system model taking into account the relaxation of the qubit.

lengths. We find the average gate error bottoms out at 0.020 ± 0.007, at σ = 3 ns. However,
simple theory (solid line) taking into account the T1 and T2∗ for a two-level system suggests
further improvement for shorter gate lengths and predicts an average gate error of 0.007 for
σ = 1 ns.
The error mechanism for these shortest pulses can be inferred by looking at ⟨σz ⟩ of the
qubit for a certain combination of gates. Figure 6.18 shows the measurement of ⟨σz ⟩ for all
concatenations of a ±π rotation along either x or y with a ±π/2 rotation along either x or y.
The calibrated levels of ⟨σz ⟩ = −1 and ⟨σz ⟩ = 1 are obtained from two separate experiments
where no single-qubit gate is applied and when a π pulse with σ = 6 ns is used. Here the
longer π pulse is used to calibrate the scale because of the increased prevalence of errors
for shorter pulses. The theoretical results of all the combinations of pulses correspond to
⟨σz ⟩ ∈ {−1, 0, 1}. However, we can see in figure 6.18 that many experiments (red-outlined
162 initialization and benchmarking

1.0

0.5

0.0
⟨σz ⟩

-0.5

-1.0
R x (π)

R x (−π)R x (−π)

R x (π)R x (π/)
R y (π)R y (π)

R x (π)R y (−π/)
R y (−π/)
R y (−π)

R y (π/)

R x (π)R y (π)

R y (π)R x (−π/)
R x (π/)

R y (π)R y (−π/)
R x (−π)

R x (π)R y (−π)

R x (π)R y (π/)
R y (π)R y (−π)

R x (π)R x (−π/)
R y (π)R x (−π)

R y (π/)R x (−π/)
R y (π)

R y (π)R y (π/)
1

R y (π)R x (π/)
R y (π)R x (π)
R x (−π/)

R x (π/)R x (π)

R y (π/)R x (π)

R y (π/)R y (π/)
R x (π)R x (π)

R x (π/)R x (−π/)

R y (π/)R x (−π/)
R x (π/)R y (π/)
R x (π/)R y (π)

R x (π/)R y (−π/)
R y (π/)R y (π)
R x (π/)R x (π/)

R y (π/)R x (π/)
Figure 6.18: Composite x and y rotations with standard Gaussian pulse shaping. Inferred
⟨σz ⟩ for a set of simple qubit rotation experiments. All combinations of rotations should result
in the qubit giving ⟨σz ⟩ = −1, 0, 1, as indicated by the blue shaded bars. The experimentally
determined values are given as the red-outlined bars. Large errors are found in composite pulse
sequences involving both x and y rotations. The first five experiments are used for calibration.

bars) deviate from their theoretical expectations (solid blue bars). Specifically, the greatest
errors occur for combinations of gates where the second gate is in the opposite quadrature
from the first. For example, the sequence R x (π)R y (π/2) should place the qubit in an equal
superposition state of ∣0⟩ and ∣1⟩ such that ⟨σz ⟩ = 0. But in the experiment, we find this to
give ⟨σz ⟩ = −0.87, which is almost an order unity error. The symmetry of the measured ⟨σz ⟩
around 0 for the experiments R x (π)R y (π/2) and R x (π)R y (−π/2) (which gives ⟨σx ⟩ = 0.44)
suggests a phase error which has resulted before the second rotation in each case, R y (±π/2).
The second gate thus performs a rotation around an axis not commensurate with the one
perpendicular to the x rotation axis defined by the first pulse.

6.4.3 Experimentally implementing derivative pulse shaping

The critical issue for this sample is the larger 01 and the fact that we operate the transmon
above the cavity, with the higher order transitions actually closer to the cavity resonance
in frequency than ω01 . The reduced detuning to the transition frequencies ω12 and ω02 /2
combined with the larger coupling strength 01 suggest that direct coupling to the ∣2⟩ state of
the transmon might not be negligible for shorter pulses that have higher frequency bandwidth.
 6.4. derivative-based pulse shaping 163

(a)

10

-10
Amplitude [arb]

(b)

10

-10
Q

0 2 4 6 8 10
Time [ns]
Figure 6.19: Gaussian and derivative on I and Q quadratures. (a) A standard truncated
Gaussian is applied on one quadrature. (b) Frequency modulated truncated derivative of
Gaussian is applied on the other quadrature. The pulse shape is shown in the lower left corner.
164 initialization and benchmarking

1.0

0.5

0.0
⟨σz ⟩

-0.5

-1.0
R x (π)

R x (−π)R x (−π)

R x (π)R x (π/)
R y (π)R y (π)

R x (π)R y (−π/)
R y (−π/)
R y (−π)

R y (π/)

R x (π)R y (π)

R y (π)R x (−π/)
R x (π/)

R y (π)R y (−π/)
R x (−π)

R x (π)R y (−π)

R x (π)R y (π/)
R y (π)R y (−π)

R x (π)R x (−π/)
R y (π)R x (−π)

R y (π/)R x (−π/)
R y (π)

R y (π)R y (π/)
1

R y (π)R x (π/)
R y (π)R x (π)
R x (−π/)

R x (π/)R x (π)

R y (π/)R x (π)

R y (π/)R y (π/)
R x (π)R x (π)

R x (π/)R x (−π/)

R y (π/)R x (−π/)
R x (π/)R y (π/)
R x (π/)R y (π)

R x (π/)R y (−π/)
R y (π/)R y (π)
R x (π/)R x (π/)

R y (π/)R x (π/)
Figure 6.20: Composite x and y rotations with derivative pulse shaping. Inferred ⟨σz ⟩ for a
set of simple qubit rotation experiments. All combinations of rotations should result in the
qubit giving ⟨σz ⟩ = −1 0 1, as indicated by the blue shaded bars. The first five experiments from
the left are used for measurement calibration. The experimentally determined values are shown
as the red-outlined bars in good agreement with the theoretical expectation.

By employing pulse shaping based on the optimal control technique of DRAG (sec-
tion 4.2.3), we are able to reduce the single-qubit gate errors at lower pulse widths. To recall,
the DRAG [90] technique is a protocol for pulse shaping such as to reduce the error caused
by the presence of a third level.
Here, we experimentally implement derivative pulse-shaping (DPS) and show the im-
provement of single-qubit gate performance. In DPS, when a single-qubit rotation pulse
is applied along the x axis, a derivative of the pulse is applied along the y axis to cancel
out the higher level leakage, and vice versa. As before, rotations around the x and y axes
are performed with in-phase and quadrature microwaves tuned to the qubit ground to first
excited state frequency, and are shaped with Gaussian envelopes, truncated to two standard
deviations σ on each side. After each gate, we still include a 5 ns buffer to avoid any overlap
with the following gate. This makes the total gate time t  = 4σ + 5 ns. The leakage due to an
x-rotation or in-phase pulse is reduced by applying a complementary quadrature tone shaped
by an amplitude scale factor D of the truncated derivative-of-Gaussian envelope. Similarly,
for a y-rotation or quadrature pulse, the same scale factor D of the truncated derivative is
applied on the x or in-phase channel.
We can visualize these truncated Gaussians and their corresponding derivatives in fig-
 6.4. derivative-based pulse shaping 165

1.0
0.9 σ = 6 ns (a) σ = 3 ns (b)
0.8
0.7
Fidelity = 1-P Error

0.6
0.5
1.0
0.9 σ= 1 ns (d) σ = 2 ns (c)
0.8
0.7
0.6
0.5
20 40 60 80 20 40 60 80
Number of Computational Gates

Figure 6.21: Randomized benchmarking for various pulse widths using derivative pulse
shaping. Total average fidelity and averaging just over the Pauli randomizations for σ =
6, 3, 2, 1 ns having implemented derivative pulse shaping. The error per gate decreases monot-
onically with decreasing σ.

ure 6.19, here with the scaling parameter D = 1. In practice, this scale factor D is tuned
via a simple calibration experiment. The calibration is an iterative procedure, starting with
standard Gaussians on one quadrature and D = 0, and measuring the homodyne voltage
for applying 1, R x (π), R x (π/2),R x (π)R y (π/2), R x (π)R y (−π/2). In this case, we are able
to obtain the homodyne voltage levels corresponding to being in ∣0⟩, ∣1⟩, and the equal su-
perposition of the two basis states. The last two experiments, as discussed previously and
part of the experiment shown in figure 6.18, contain a systematic symmetric error around
the result for R x (π/2). Next, we repeat the same experiment but program the pulses such
that D = 0.1 and therefore microwaves are applied both to x and y. Ideally, the results of the
last three pulse sequences, R x (π/2),R x (π)R y (π/2), R x (π)R y (−π/2) should be the same. As
D is scaled up, the three values approach one another. If D is increased too much, the last
two experiments give values which deviate from the superposition level again. Therefore,
from this calibration technique the scaling of the derivative pulse on the complementary
quadrature can be determined.
For the qubit discussed in this section, the level D is found to be 0.4. We can first explore
166 initialization and benchmarking

the collection of gates where we concatenate x rotations with y rotations. Figure 6.20 shows
the same experiment as in figure 6.18, but having implemented the DPS on the conjugate
quadrature. All of the previous errors are now removed and we have excellent agreement
with theory.
What is now the effect of these pulses on the average gate fidelity? We repeat the ran-
domized benchmarking protocols for various pulse widths σ. The fidelity as a function of
number of gates averaging over the 8 different Pauli randomizations is shown in figure 6.21.
The 32 individual (recall 8 Pauli and 4 Clifford randomizations) randomized traces along
with the average fidelity as a function of the number of gates for a number of σ are shown
in figure 6.22. The clearest and most striking improvement can be seen for σ = 1, 2 ns. The
spread of the 32 unique traces, with order unity deviations for σ = 1 ns without DPS, is now
improved considerably.
Next, we extract the average gate fidelity from the exponential fits and plot it against the
pulse width in figure 6.23, where we have included the pre-DPS results as well as the simple
decoherence theory. Using DPS, we obtain a minimum error per gate of 0.007 ± 0.006 for our
shortest possible gates, with σ = 1 ns. As we can see from figure 6.23, we experience improved
gate performance for σ = 1 − 3 ns which agrees remarkably well with the two-level system
theory.

6.4.4 Summary of DPS

Thus, DPS allows us to improve gate performance and use shorter gate lengths. It seems
as though the additional error due to the finite and small anharmonicity of the transmon
can be mostly removed from the system through optimal control of the single-qubit gates.
The experiment also demonstrates that our single-qubit gates are limited by their coherence
times and the sensitivity of our hardware, as our current arbitrary waveform generators only
provide 1 ns resolution. Additional time resolution would permit even more complicated
optimal control schemes and even shorter pulses. Combined with improved coherence times,
the fault-tolerant threshold of 10−4 error per gate can become within reach. However, the
current experiment does not test explicitly whether the error mechanism is leakage to ∣2⟩. In
the future, we would like to perform experiments which characterize the population of this
level as a function of shorter gate lengths, as well as implement higher-order corrections as
proposed by Ref. [90] when the hardware control of our pulse-shaping can beat 1 ns resolution.
 6.4. derivative-based pulse shaping 167

no DPS DPS
1.0
σ= 6 ns
0.8
0.6
0.4
0.2

1.0
σ= 3 ns
0.8
0.6
0.4
Fidelity = 1-PError

0.2

1.0
σ= 2 ns
0.8
0.6
0.4
0.2

1.0
σ= 1 ns
0.8
0.6
0.4
0.2
20 40 60 80 20 40 60 80
Number of Computational Gates

Figure 6.22: Trace by trace with and without derivative pulse shaping. All unique 32 random-
ized traces with and without the derivative pulse shaping for σ = 6, 3, 2, 1 ns. The spread in the
traces can be seen to get visibly worse without the shaping, and visibly better with the shaping.
168 initialization and benchmarking

Pulse Width σ (ns)

1 2 3 4 6

0.1
6
4
Average Error Per Gate

0.01
6
4 RB with gaussians
RB with DPS
2 simple T 1 theor y
0.001
10 15 20 25
Total Gate Length (ns)
Figure 6.23: Error per gate with derivative pulse shaping. Extracted error per gate versus
total gate length with (blue square markers) and without (red square markers) the derivative
pulse shaping. We find that with the derivative shaping, an excellent agreement with the simple
theory (black solid line). The minimum error per gate of 0.007 ± 0.006 is reached at σ = 1 ns.

6.5 Chapter summary

In this chapter, we have demonstrated the critical starting requirements for a quantum infor-
mation processor. Specifically, for the circuit QED system with transmon qubits, we have
observed a lower bound on the average photon number in the system of 0.003, corresponding
to an initial ground state polarization of the qubit of over 0.9999. Furthermore, through
various gate error characterization techniques combined with optimized pulse-shaping pro-
tocols, we have benchmarked single-qubit gates to ∼ 1% errors. While initialization and
single-qubit operations are necessary, they are also not sufficient for quantum computing.
The next chapters will explore the expansion of the circuit QED architecture to two qubits and
the prospects of the cavity bus for coupling and generating entanglement will be determined.
 CHAPTER 7

Two-Qubit Circuit QED: Riding the Quantum Bus

W
ith universality of quantum computing (section 2.1) dictating the need for a two-qubit
entangling gate, we now expand the circuit QED architecture from a robust single-
qubit system to one with two qubits. Although single-qubit gates are now ubiquitous across
most superconducting qubit implementations, operations on and the coupling of multiple
qubits are still a subject of ongoing research. Specifically, for flux qubits, two-qubit coupling
[128] and a controllable coupling mechanism have been realized [70, 72, 129]. Two phase
qubits have also been successfully coupled [130] and the entanglement between them has
been observed [67]. However, all of these interactions have been realized by connecting
qubits via lumped circuit elements (section 3.2), with capacitive coupling (section 3.2.1) in
the case of charge and phase qubits, and inductive coupling (section 3.2.2) for flux qubits.
Therefore, these coupling mechanisms have been restricted to local interactions and couple
only nearest neighbor qubits.
Performing gates between an arbitrary pair of distant qubits is highly desirable for a
scaleable quantum computer architecture. An efficient way to achieve this goal is to couple
the qubits to a quantum bus (section 3.2.3), which distributes quantum information among
the qubits. The primary requirement for a quantum bus architecture is to have a quantum
degree of freedom which can interact strongly with independent quantum systems for storage
or transfer of information. There are several physical systems in which one could realize a

169
170 two-qubit circuit qed

quantum bus. A particular example is trapped ions [13, 14] in which a variety of quantum
operations and algorithms have been performed using the quantized motion of the ions
(phonons) as the bus. Photons are another natural candidate as a carrier of quantum informa-
tion [131, 132], because they are highly coherent and can mediate interactions between distant
objects. To create a photon bus, it is helpful to utilize the increased interaction strength
provided by the techniques of cavity quantum electrodynamics (section 3.3), where an atom is
coupled to a single cavity mode. In the strong coupling limit (section 3.3.1) the interaction is
coherent, permitting the transfer of quantum information between the atom and the photon.
Such a photon bus has led to the generation of entanglement between atoms using a Rydberg
atom cavity QED [133–135] experiment.
Circuit QED as previously described (section 3.4) is a realization of the physics of cavity
QED with superconducting qubits and a microwave cavity on a chip. Here in this chapter,
we will show the first implementation of a quantum bus in circuit QED, using microwave
photons confined in a transmission line cavity, to couple two superconducting qubits on
opposite sides of a chip. Instead of providing a lumped element coupling, the cavity behaves
as a distributed circuit element permitting long-range quantum interaction which need not
be nearest-neighbor. Section 7.2 will detail the coupling of two qubits to the microwave
cavity. A two-qubit interaction (section 7.2.1) is mediated by the exchange of virtual rather
than real photons over the bus, with the added benefit of avoiding direct loss through the
cavity. Then, in section 7.4 we show the ability to use an ac-Stark interaction for fast control
of the qubits to switch the coupling effectively on and off. Controlling both the individual
qubits independently as well as the coupling interaction, we demonstrate coherent transfer of
quantum states between the qubits. The same cavity which couples the qubits is also used to
perform multiplexed control and measurement of the qubit states. The experiment presented
in this chapter is a more detailed description of that described in Ref. [136] and reflects the
combined efforts by myself and postdoc Johannes Majer. These results represent the first step
for circuit QED in the direction of generating and detecting entanglement, and will serve as
a useful springboard to the experiments presented in chapters 8 and 9 to immediately follow.

7.1 Experimental details

The cavity bus described in this chapter will refer to sample cQED157, with fabrication details
given in chapter 5. The sample and basic experimental schematic are shown in figure 7.1. The
two superconducting transmon qubits are 5 mm apart at opposite ends of the superconducting
 7.1. experimental details 

(a)

(b)

7mm

300mm 100mm

Figure 7.1: Sample and scheme used to couple two qubits to an on-chip microwave cavity.
Circuit (a) and optical micrograph (b) of the sample, cQED157, with two transmon qubits
coupled by a microwave cavity. The cavity is formed by a coplanar waveguide (light blue)
interrupted by two coupling capacitors (purple). The resonant frequency of the cavity is
ωC /π = . GHz and its width is κ/π =  MHz. The output coupling capacitor is shown in
the purple inset. The cavity is operated as a half-wave resonator (L = λ/ = . mm) and the
electric field in the cavity is indicated by the gray line. The two transmon qubits are located
at opposite ends of the cavity where the electric field has an antinode. Each transmon qubit
consists of two superconducting islands connected by a pair of Josephson junctions and an
extra shunting capacitor (interdigitated finger structure in the green inset). The microwave
signals enter the chip from the left, and the response of the cavity is amplified and measured
on the right.

coplanar waveguide resonator (ωC /π = . GHz, κ/π =  MHz). Recalling the transition
frequency of the transmon qubits from (3.17), the split-pair of Josephson junctions give an
external flux tunable Josephson energy, EJ = EJmax ∣cos(π Φ̃/Φ )∣. The external magnetic flux
Φ̃ is applied with a superconducting magnet in the cryostat to tune both qubit transition
energies. Recall that for this sample, we designed the size of the two loops to be different
and incommensurate by a factor of approximately /, so that control of the two transition
frequencies is attainable with a certain degree of independence. The left qubit (qubit , color
coded green) has a charging energy of EC /h =  MHz and maximum Josephson energy
172 two-qubit circuit qed

of EJ1max /h = 14.9 GHz. The right qubit (qubit 2, color coded red) has a charging energy of
EC2 /h = 442 MHz and maximum Josephson energy of EJ2 max
/h = 18.9 GHz.

7.2 Two-qubit spectroscopy

We can demonstrate strong coupling of each of the qubits separately to the cavity bus by
varying the externally applied magnetic flux, until each of the two qubits are tuned into
resonance with the cavity as shown in figure 7.2. This experiment involves applying a single
microwave excitation tone continuously, and we monitor the transmitted homodyne voltage
amplitude while sweeping the frequency of the applied tone. With both of the qubits not
excited, we see only a single peak in the measured homodyne signal, following a Lorentzian
lineshape centered at the cavity frequency. However, the qubit transition frequencies can
be tuned into resonance with the cavity using the external magnetic flux. In figure 7.2, we
observe vacuum Rabi splittings (section 3.3.1) of both qubits with the cavity, indicating
that each qubit can in fact reach the strong coupling limit with the cavity. Theoretically
determined frequencies for the left and right qubit are shown in the green and red dashed
lines, respectively. The experimentally determined frequencies follow the theoretical flux
dependence of the Josephson energy and allows us to extract the magnetic field corresponding
to Φ0 for each of the qubits. In the vacuum Rabi splitting, each of the peaks corresponds to a
superposition of qubit excitation and a cavity photon in which the energy is shared between
the two systems. Furthermore, from the frequency difference at the maximal splitting for
each qubit, the coupling parameters (1),(2) /π ≈ 105 MHz can be determined.
In the dispersive limit, both qubits are detuned from the resonator such that

∣∆(1),(2) ∣ = ∣ω(1),(2) − ωC ∣ ≫ (1),(2) . (7.1)

Recall from chapter 4, that in this limit, we can use second order perturbation theory to
describe the full system with the two qubits and the cavity with the effective Hamiltonian:

ħω(1) (1) ħω(2) (2)

Heff = σz + σz + ħ (ωC + χ(1) σz + χ(2) σz ) a † a
(1) (2)
2 2 (7.2)
+ ħJ (σ− σ+ + σ− σ+ ) .
(1) (2) (2) (1)

In this regime, no energy is exchanged with the cavity. However, the qubits and cavity are still
dispersively coupled, resulting in a qubit-state-dependent shift ±χ(1),(2) of the cavity frequency
 7.2. two-qubit spectroscopy 173

5.4
Frequency (GHz)

5.3

5.2

5.1

5.0
-0.4 -0.2 0.0 0.2 0.4
Magnetic Field (Gauss)
Figure 7.2: Strong coupling of two superconducting qubits. Density plots of the transmission
as a function of drive frequency and magnetic field. Blue (red) indicates low (high) transmission.
When the qubits are far detuned from the cavity frequency ωC = 5.22 GHz, there is a single
peak in transmission, as is seen in at −0.1 Gauss. However, by tuning the global external flux,
the qubit frequencies can be tuned into resonance with the cavity, and we can observe avoided
crossings with the cavity. When exactly on resonance, the cavity peak is split into two vacuum
Rabi peaks, from which the qubit-cavity coupling strengths are determined.

∣, ⟩ ∣, ⟩ ∣, ⟩ ∣, ⟩

1.0
Transmission

0.5

0.0
5.18 5.20 5.22 5.24 5.26
Frequency (GHz)
Figure 7.3: Dispersive shifts of the cavity. In the dispersive regime, the bare cavity transmission
is shifted to four frequencies depending on the state of the two qubits (∣1, 1⟩ in blue, ∣0, 1⟩ in
red, ∣1, 0⟩ in green, ∣0, 0⟩ in black). Here, we show simulated Lorentzians with κ/2π = 33 MHz
and χ(1) = 7 MHz and χ(2) = 5 MHz. Measurements are generally performed by looking at
transmission corresponding to the ∣0, 0⟩ in black.
174 two-qubit circuit qed

∣, ⟩ , n = 

ω + ω
∣, ⟩ , n =  ∣, ⟩ , n = 

∆
() () ∣, ⟩ , n = 
ω ω

ωC
∣, ⟩ , n = 

Figure 7.4: Scheme of the virtual photon swap interaction. When the qubits are detuned from
the cavity (∣∆(1),(2) ∣ =≫ (1),(2) ) the qubits both dispersively shift the cavity. The excited state
in the left qubit ∣10⟩ ⊗ ∣n = 0⟩ interacts with the excited state in the right qubit ∣01⟩ ⊗ ∣n = 0⟩
via the exchange of a virtual photon ∣00⟩ ⊗ ∣n = 1⟩ in the cavity.

(see figure 7.3) or equivalently an ac Stark shift of the qubit frequencies (section 4.2.4). The
frequency shift χ(1),(2) can be calculated from the detuning ∆(1),(2) and the measured coupling
strength (1),(2) (3.45). The last term describes the interaction between the qubits, which is a
transverse exchange interaction of strength J = (1) (2) (1/∆(1) + 1/∆(2) )/2. The qubit-qubit
interaction (section 4.3.2) is a result of virtual exchange of photons with the cavity. When the
qubits are degenerate with each other, an excitation in one qubit can be transferred to the other
qubit by virtually becoming a photon in the cavity (see figure 7.4). However, when the qubits
are non-degenerate ∣ω(1) − ω(2) ∣ ≫ J this process does not conserve energy, and therefore
the interaction is effectively turned off. Thus, instead of modifying the actual coupling
constant [70, 72, 129], we control the effective coupling strength by tuning the qubit transition
frequencies. This is possible since the qubit-qubit coupling is transverse (section 2.3.3),
which also distinguishes our experiment from the situation in liquid-state NMR quantum
computation, where an effective switching-off can only be achieved by repeatedly applying
decoupling pulses [137].
We can observe the coherent interaction between the two qubits via the cavity by per-
forming spectroscopy of their transition frequencies (see figure 7.5). Spectroscopy is a dual
microwave tone experiment. A measurement tone is applied continuously at the cavity trans-
mission corresponding to both qubits being in their ground states, while a second tone is
swept in frequency away from the cavity frequency to probe for the qubit transitions. As long
as the qubit is in the dispersive regime, there is a dispersive cavity shift which depends on
the state of the qubit (section 3.4.2). If the probe microwave signal is resonant with the qubit
 7.2. two-qubit spectroscopy 175

transition, the qubit state will be driven to a mixed state of ∣0⟩ and ∣1⟩. This is the result of
the probe signal being on continuously and the T1 relaxation process of the qubit when it is
in the excited state. Therefore, when the drive is resonant with the qubit, the transmission
at the cavity transmission frequency ωC is reduced due to the state of the qubit, shifting
the transmission to the frequency ωC + χ. This homodyne voltage is detected and can be
displayed as a function of the drive frequency to produce spectroscopy maps of the qubit
transition frequencies while also varying the applied external magnetic flux.

7.2.1 Qubit-qubit avoided crossing

Varying the external magnetic flux Φ̃, we spectroscopically observe a map of the two qubit
transitions, as shown in figure 7.5. We can overlay the theoretical calculation for the transition
frequencies (3.17). By varying Φ̃, we find that it is possible to tune the qubits through
resonance with each other (see figure 7.5b), revealing an avoided crossing. This reflects the
virtual-photon swap interaction (section 4.3.2). The magnitude of the splitting agrees well
with the theoretical value 2J = 2(1) (2) /∆ = 2π ⋅ 26 MHz when one takes into account that
(1),(2) vary with frequency for a transmon qubit [61]. The splitting is well resolved, with a
magnitude J much greater than the qubit line widths, indicating both a coherent coupling
and that the qubits are in the strong dispersive limit [110]. Note that although the coupling
strength J is smaller than the cavity decay rate κ/2π ∼ 33 MHz, the avoided crossing is
nearly unaffected by the cavity loss. This is possible in such a large-κ cavity, required for fast
measurements, because only virtual photons are exchanged; if real photons were used, the
cavity induced relaxation of the qubits (see Purcell effect section 3.5.1 and [82]) would make
coherent state transfer unfeasible.

7.2.2 The dark state

Another manifestation of the coherence of this interaction is the observation of a dark state.
There is a disappearance of one of the spectroscopy peaks near the avoided crossing point,
which is due to destructive interference associated with the fact that the qubits are separated
by half a wavelength. This effect is a result of the physical location of the qubits, being on
opposite ends of the λ/2 microwave cavity. Recall from that an applied microwave drive tone
takes the form

Hdrive = (a + a † ) (ξe −iωd t + ξ∗ e iωd t ) , (7.3)

176 two-qubit circuit qed

8.0
(a)
7.5
Frequency (GHz)

7.0
6.5
6.0
5.5
5.0
0.25 0.30 0.35 0.40 0.45 0.50
6.6
(b) ∣⟩ − ∣⟩ (c)
Frequency (GHz)

∣⟩

6.5 Dark State

∣⟩ + ∣⟩ ∣⟩

0.460 0.465 0.460 0.465

Magnetic Field (Gauss)
Figure 7.5: Two qubit spectroscopy. Cavity transmission and spectroscopy of single and
coupled qubits. (a) The transmission through the cavity as a function of applied magnetic field
is shown in the frequency range between 5 GHz and 5.4 GHz. When either of the qubits is in
resonance with the cavity, the cavity transmission shows an avoided crossing due to the vacuum
Rabi splitting. Above 5.5 GHz, spectroscopic measurements of the two qubit transitions is
shown. The density plot reflects the change in transmission of a probe signal on resonance
with the cavity transmission when both qubits are in the ground state. The dashed lines show
the resonance frequencies
√ of the two qubits, which are a function of the applied flux according
(1),(2)
to ω(1),(2) = ωmax ∣cos(π Φ̃/Φ0 )∣. The maximum transition frequency for the first qubit
(1) (2)
is ωmax /2π = 7.8 GHz and for the second qubit is ωmax /2π = 6.45 GHz. For strong drive
powers, additional resonances between higher qubit levels are visible. (b) Spectroscopy of the
two-qubit crossing. The qubit levels show a clear avoided crossing with a minimal distance
of 2J/2π = 26 MHz. At the crossing the eigenstates of the system are symmetric and anti-
symmetric superpositions of the two qubit states. The spectroscopic drive is anti-symmetric
and therefore unable to drive any transitions to the symmetric state, resulting in a dark state.
(c) Predicted spectroscopy at the qubit-qubit crossing using a Markovian master equation that
takes into account higher modes of the cavity.
 7.2. two-qubit spectroscopy 177

where the ωd is the drive frequency and ξ is the drive strength. Hidden in this drive Hamilton-
ian is the voltage of the cavity, V0 (a + a † ), where the V0 has been absorbed into ξ. This is an
important subtlety however, that the drive is in fact a capacitive coupling between the voltage
mode of the cavity and that of a drive cavity which remains in a highly-excited coherent
state [62]. For the particular arrangement of the qubits being located on opposite ends of the
sample, the voltage which couples to the two qubits will be different. Specifically, since we
drive the cavity with a λ/2 mode, the electric field at the different ends will have opposite
signs. Therefore, the voltage seen by the left qubit near the input port will be V0 (a + a† ),
whereas the voltage at the right qubit near the output port will be −V0 (a + a † ). This extra
negative sign means that the drive (with similar treatment of assuming RWA and dispersive
limit as in section 7.2.2) will now take the form

2(1) ξ (2) 2
(2) ξ
Hdrive = σx −
(1)
σ x . (7.4)
ω(1) − ωd ω(2) − ωd
When performing spectroscopy, both qubits start out in the ground state, ∣00⟩. The action of
this drive on the ground state will then be

⎛ (1) (2) ⎞
Hdrive ∣0, 0⟩ = 2ξ ∣1, 0⟩ − ∣0, 1⟩ , (7.5)
⎝ ∆d
(1) (2)
∆d ⎠

are the detunings for the qubits from the drive ω(1),(2) − ωd .
(1),(2)
where ∆d
At the avoided crossing, the eigenstates are superpositions of the single qubit states. In
√
particular, the state with lower frequency is the symmetric triplet state ∣+⟩ = (∣0, 1⟩+∣1, 0⟩)/ 2
√
and the state at higher frequency is the antisymmetric singlet state ∣−⟩ = (∣0, 1⟩ − ∣1, 0⟩)/ 2.
Therefore, when tuning to the avoided crossing, we can drive the two qubits with Hdrive and
compute the overlap with ∣±⟩,

(1) (2)
⟨ 0, 1 + 1, 0 ∣ Hdrive ∣ 0, 0 ⟩ = − (7.6a)
∆(1) ∆(2)
(1) (2)
⟨ 0, 1 − 1, 0 ∣ Hdrive ∣ 0, 0 ⟩ = (1) + (2) . (7.6b)
∆ ∆
For (1) = (2) , the symmetric state will be ‘dark’ when the two qubits are in resonance with
each other in the sense that the drive will not make real transitions to the state. Moreover, for
this state to not couple to the drive, means that it is in fact protected against decay through
the cavity. Conversely, the decay from the anti-symmetric state is enhanced, similar to
178 two-qubit circuit qed

super-radiant effects observed in atomic physics [138, 139]. Figure 7.5c shows the simulated
spectroscopy at the qubit-qubit crossing, which reproduces all qualitative features of the
measured data. The simulation is performed via a Markovian master equation which takes
into account higher modes of the cavity and uses parameters such as (1),(2) obtained from
the vacuum Rabi data, κ, and coherence times T1 and T2 for both qubits near the resonance
point.
The presence of the dark state thus reflects a spectroscopic verification of the coherent
virtual-photon coupling of the two qubits. However, it would be even better to verify the
coupling through time-domain experiments and observe the coherent swapping of states
between the qubits. Yet to be able to perform that experiment, we need to lay the groundwork
for the qubit readout.

7.3 Multiplexed joint qubit readout

In addition to acting as a quantum bus, the same cavity is also used for multiplexed readout
and control of the two qubits. Here, “multiplexed” refers to acquisition of information or
control of more than one qubit via a single channel.
To address the qubits independently, the flux is tuned such that the qubit frequencies
are 88 MHz apart (ω(1) = 6.617 GHz, ω(2) = 6.529 GHz), making the qubit-qubit coupling
negligible. Rabi driving experiments showing individual control are performed by applying
an rf-pulse at the resonant frequency of either qubit, followed by a measurement pulse at
the resonator frequency. The measured homodyne amplitude response (see figure 7.6 for
driving each qubit is consistent with that of a single qubit oscillation and shows no beating,
indicating that the coupling does not affect single-qubit operations and readout.
With similar measurements the relaxation times (T1 ) of the two qubits are determined
to be 78 ns and 120 ns, and with Ramsey fringe measurements the coherence times (T2 ) are
found to be 120 ns and 160 ns. The T1 times are consistent with the Purcell effect, as the cavity
is relatively fast decaying with κ/2π = 33 MHz.
The ability to simultaneously readout the states of both qubits using a single line is
demonstrated by measuring the cavity phase shift, proportional to χ(1) σz + χ(2) σz , after
(1) (2)

applying a π-pulse to one or both of the qubits. Figure 7.7 shows the response of the cavity
after a π-pulse has been applied on the first qubit (green points), on the second qubit (red
points) or on both qubits (blue points). For comparison the response of the cavity without
any pulse applied (black points) is shown. Since the cavity frequency shifts for the two qubits
 7.3. multiplexed joint qubit readout 179

(a)
Rabi pulse measurement
∆t pulse

(b) (c)
0.5

0.0
Polarization

-0.5
( ) ( )

-1.0
0 50 100 150 0 50 100 150
Rabi Pulse Length (ns)

Figure 7.6: Independent Rabi driving of two qubits. (a) Pulse protocol for performing Rabi
oscillations. The Rabi pulse is applied resonant to the qubit transition frequency for a varying
duration of ∆t. The response of the cavity transmission is measured after the pulse is turned off.
Oscillations of quadrature voltages are measured for each of the qubits and mapped onto the
z
polarization ⟨σ1,2 ⟩. (b–c) Rabi oscillations of qubit 1 and qubit 2. The solid line shows results
from a master equation simulation, which takes into account the full dynamics of the two
qubits and the cavity. The absence of beating in both traces is a signature of the suppression of
the qubit-qubit coupling at this detuning.

are different (χ(1) ≠ χ(2) ), we are able to distinguish the four states ∣00⟩, ∣01⟩, ∣10⟩, and ∣11⟩ of
the qubits with a single readout line. Although not performed in this experiment due to the
relaxation limited signal-to-noise ratio, the joint readout can be combined with single-qubit
rotations to give a full reconstruction of the density matrix (state tomography as described in
section 2.5.2).
The solid lines in figure 7.7 show the results from a theoretical calculation taking into
account the full dynamics of the cavity and the two qubits, including the relaxation rates of
the qubits. The agreement of the theory with the measured response shows that the measured
contrast is the maximum expected. From the calculated values one can estimate the selectivity,
i.e. the ability to address one qubit without affecting the other, S = (Pa − Pu )/(Pa + Pu ), where
Pa and Pu are the maximum populations in the excited state of the addressed qubit and in
the excited state of the unaddressed qubit, respectively. The selectivity for qubit 1 is 87 % and
qubit 2 is 94 %, which indicates good individual control of the qubits.
180 two-qubit circuit qed

meas.
6
∣, ⟩

Homodyne Voltage (mV)

∣, ⟩
4
π
∣, ⟩

2 ∣, ⟩
π

0
π
π -100 0 100 200 300 400
Time (ns)
Figure 7.7: Two qubit multiplexed readout. Pulse schemes shown on the left for preparing the
four different states and then performing a homodyne measurement. The homodyne response
(average of 1,000,000 traces) of the cavity after a π pulse on qubit 1 (green), qubit 2 (red), and
both qubits (blue). The black trace shows the level when no pulses are applied. The contrasts
(i.e. the amplitude of the pulse relative to its ideal maximum value) for these pulses are 60%
(green), 61% (green) and 65% (blue). The solid line shows the simulated value including the
qubit relaxation and the turn-on time of the cavity. The agreement between the theoretical
prediction and the data indicates the measured contrast is the maximum observable. From
the theoretical calculation one can estimate the selectivity for each π-pulse to be 87% (qubit 1)
and 94% (qubit 2). This figure of merit is not at all intrinsic and that it could be improved by
increasing the detuning between the two qubits for instance, or using shaped excitation pulses.

7.4 Coherent state transfer: Stark swap

Although the spectroscopy (figure 7.5b) suggests a coherent qubit-qubit interaction through
the avoided crossing and the presence of the dark state, we would like to observe the inter-
action in the time-domain. We can perform coherent state transfer, or qubit state swap, in
the time-domain if we can turn the effective qubit-qubit coupling on and off, rapidly, on
time-scales corresponding to the decoherence times of the two qubits.
The simplest protocol to think of is to use the external magnetic flux to pulse into the
avoided crossing of figure 7.5b. However, in this implementation of the experiment, the flux
tunability is achieved through ramping a common superconducting coil within the cryostat.
The time-constant for ramping such a large magnet is too long to drive coherent interactions
and achieve single-qubit operations before the qubits decohere.
 7.4. coherent state transfer: stark swap 181

1.6
2
1.4
Stark Tone Power (mW)
1.2

Number of Photons
1.0
0.8 1
0.6
0.4
0.2
0.0 0
6.35 6.40 6.45 6.50 6.55
Frequency (GHz)
Figure 7.8: Two qubit Stark shift spectroscopy. Spectroscopy of qubits versus applied Stark
tone power. Taking into account an attenuation of 67 dB before the cavity and the filtering effect
of the cavity, 0.77 mW corresponds to an average of one photon in the resonator. The qubit
transition frequencies (starting at ω(1) /2π = 6.469GHz and ω(2) /2π = 6.546GHz) are brought
into resonance with a Stark pulse applied at 6.675 GHz. An avoided crossing is observed with
one of the qubit transition levels becoming dark as in figure 7.5b.

As a result, rather than the slow flux tuning, we now make use of a strongly detuned
rf-drive[91], which results in an off-resonant Stark shift (section 4.2.4) of the qubit frequencies
on the nanosecond time scale. To see how this works, consider re-writing the dispersive
two-qubit Hamiltonian as

ħ ħ
Heff = (ω(1) + χ(1) a† a)σz + (ω(2) + χ(2) a† a)σz + ħωC a † a
(1) (2)
2 2 (7.7)
+ ħJ (σ− σ+ + σ− σ+ ) .
(1) (2) (2) (1)

Although details of the ac Stark effect are given previously in section 4.3.2, here with a quick
glance at the Hamiltonian, we see that the an applied drive changes the number of photons
and thus shifts the effective qubit frequencies.
We can perform a spectroscopy experiment starting with the two qubits separated in
frequency and observe the shift of the lines as a function of increasing the power of an
off-resonant Stark drive. With the qubits originally tuned to 6.47 GHz and 6.55 GHz, and
placing a drive tone at 6.675 GHz, we spectroscopically detect the Stark shift of both qubits
182 two-qubit circuit qed

∣, ⟩

∣, ⟩

J
∣, ⟩

∣, ⟩

ω() ≠ ω() ω() = ω() ω() ≠ ω()

Bring qubits into resonance Measure in

Excite one qubit qubits swap uncoupled basis
Figure 7.9: Coherent swap protocol. Three step process for performing coherent swaps.

figure 7.8. The qubit frequencies are pushed into resonance and a similar avoided crossing is
observed as in figure 7.5. The avoided crossing of the two qubits is possible even though the
cavity couplings of each qubit, (1) and (2) , are equal as a result of the difference in detuning
between the qubit transition frequencies from the Stark drive frequency.
With the Stark drive’s ability to quickly tune the qubits into resonance, it is possible to
observe coherent oscillations between the qubits, using the following protocol (see figure 7.9):

1. Initially the qubits are 80 MHz detuned from each other, where their effective coupling
is small, and they are allowed to relax to the ground state ∣0, 0⟩.

2. Next, a π-pulse is applied to one of the qubits to either create the state ∣1, 0⟩ or ∣0, 1⟩.

3. Then, a Stark pulse of power PAC is applied bringing the qubits into resonance for
a variable time ∆t. Since ∣1, 0⟩ and ∣0, 1⟩ are not eigenstates of the coupled system,
oscillations between these two states occur, as shown in figure 7.10.

The power of the Stark pulse PAC can be varied, mapping out the same 2J interaction
but in a time-sensitive way. The resulting oscillations for various PAC is plotted in the map
shown in figure 7.10b. Figure 7.11b shows the extracted frequency of these oscillations for
different powers PAC of the Stark pulse, which agrees with the spectroscopy measurement of
the frequency splitting observed in figure 7.8. Furthermore, we observe the anti-correlation
between the swap oscillations when initially applying the π pulse on either qubit 1 or qubit 2,
as evidenced by the red and green traces of figure 7.10. These data provide strong evidence
 7.4. coherent state transfer: stark swap 183

(a)
Stark Pulse A Measurement
π ∆t Pulse

1.2
(b)
Homodyne Voltage (mV)

1.0

0.8 ∣, ⟩

0.6 ∣, ⟩

0.4

0.2
∣, ⟩
0.0
0 20 40 60 80 100
Stark Pulse Length ∆t
Figure 7.10: Coherent state exchange. (a) Pulse protocol for the coherent state transfer using
the Stark shift. The pulse sequence consists of a Gaussian-shaped π pulse (red) on one of
the qubits at its transition frequency ω1,2 followed by a square-shaped Stark pulse (brown) of
varying duration ∆t and amplitude A detuned from the qubits, and finally a square measurement
pulse (blue) at the cavity frequency. The time between the π pulse and the measurement is
kept fixed at 130 ns. (b) Coherent state transfer between the qubits according to the protocol
above. The plot shows the measured homodyne voltage (average of 3,000,000 traces) with
the π pulse applied to qubit 1 (green dots) and to qubit 2 (red dots) as a function of the Stark
pulse length ∆t. For reference, the black dots show the signal without any π pulse applied to
either qubit. The overall increase of the signal is caused by the residual Rabi driving due to the
off-resonant Stark tone, which is also reproduced by the theory. Improved designs featuring
different coupling strengths for the individual qubits could easily avoid this effect. The thin
solid lines show the signal in the absence of a Stark pulse. Adding the background trace (black
dots) to these, we construct the curves consisting of open circles, which correctly reproduce
the upper and lower limits of the oscillating signals due to coherent state transfer.

that the oscillations are due to the coupling between the qubits and that the state of the qubits
is transferred from one to the other.
√
A quarter period of these oscillations should correspond to a iSWAP, which would
184 two-qubit circuit qed

Swap Frequency (MHz)

50

40

30

0.4 0.8 1.2 1.6

Stark Power (mW)
Figure 7.11: Stark swap frequency. By repeating the Stark swap protocol for various powers
PAC , we are able to extract the frequency of the oscillations (red points) and map out the same
avoided crossing with splitting 2J as in figure 7.8, which is shown in the solid blue curve.

be a universal gate (section 2.3.3). However, the short coherence times of this sample make
√
concatenation of a two-qubit iSWAP with single-qubit operations difficult. In order to
experimentally generate entangled states and characterize an interaction gate, better qubit
coherence is necessary.
One significant error which can be seen in the experiments shown in figure 7.10b is the
positive slope of the oscillations. The black curve represents a control experiment in which
both qubits are kept in the ground states and only the Stark pulse is turned on. The slope
is evident there as well, without any oscillations. A similar effect is in fact seen in the Stark
shift spectroscopy map of figure 7.8, as a gradient in the measured homodyne amplitude can
be seen in the background as the Stark drive power is increased. This effect is most likely
attributed to a power-dependent shift of the cavity resonance. The increased Stark drive,
despite being off-resonant from the cavity, will at large enough powers begin to Stark shift
the cavity frequency, such that the transmission of the measurement drive which is locked to
the starting cavity frequency is reduced during the experiment.
 7.5. chapter summary 185

7.5 Chapter summary

The observed qubit-qubit avoided crossing and the coherent state transfer demonstrate that
the cavity in a circuit QED system can act as a coupling bus for superconducting qubits.
The interaction is coherent and effectively switchable, as evidenced by the avoided crossing
in spectroscopy and the ability to coherently swap in the time-domain. Furthermore, the
coupling is long range and could possibly be extended to non-nearest neighbors. By operating
in the dispersive regime of cavity QED, the qubit interactions are protected against loss in
the bus by the use of virtual photons. The direct improvements which are necessary to take a
leap ahead towards generating and detecting entangled states are improved coherence times
and an improved method of turning on and off the interaction. The coherence times of
the sample studied in this experiment were multi-mode Purcell limited [82]. By reducing
the cavity linewidth, the loss of polarization via spontaneous emission would be reduced.
Furthermore, although the Stark shift is a creative way of turning the qubit-qubit interaction
on and off, a better option would be to directly tune the flux, which we can hope to achieve
with high-bandwidth on-chip flux bias. The next two chapters will detail our extension of
the experiment described in this chapter, implementing the two aforementioned changes,
and cementing the circuit QED architecture as a simple but viable quantum information
processor.
 CHAPTER 8

Entanglement On-Demand and Joint Readout

E
ntanglement, non-classical correlations between qubits, is often seen as being a critical
resource for experimental progress in quantum information science. Although its role
in generating speed-up in quantum computers over classical computers is still a subject of
theoretical debate, experimental verifications of its ‘spooky’ behavior over large distances
certainly support its importance in transmitting and storing quantum information. In the
previous chapter (chapter 7), we demonstrated the first steps towards entangling two super-
conducting qubits via a cavity bus. Furthermore, we could see the possibility to employ the
same bus which provides the interaction to act as a multiplexed readout of the two qubit
quantum state. Here, through the implementation of on-chip fast flux bias lines and improved
qubit coherence times, we are able to take both of those experimental concepts of the circuit
QED architecture a step further. First, in section 8.3 we present a new two-qubit interaction,
tunable in strength by two orders of magnitude on nanosecond time scales, which is mediated
by the cavity bus and relies on the higher excitation manifolds of the transmon qubits. Such
an interaction leads to the generation of maximally entangled states, i.e.the four canonical
Bell states (section 8.3.2).
However, the accurate and reliable detection of such quantum states and their degree of
entanglement is itself a major necessity and nontrivial problem for quantum information
systems. In any experiment one obtains information about the quantum system only through

187
188 entanglement and joint readout

the observation of the output from a detector, whose classical imperfections can introduce bias
and noise. As a result, to make precise statements about intrinsic properties of quantum states,
such as entanglement or purity (section 2.6), it is imperative to have a full understanding
of the measurement process. In traditional quantum information processing architectures,
such as those employing photons or trapped ions, the relationship between a quantum state
and the quantities measured has been well established. In addition, the fidelity of single-shot
measurements can in such cases be very high (∼ 99.99% for ions [140]) . Consequently, the
difficulties of calibration are minimized and the paradigm for correlation measurements [15,
16] is to record coincidences between individual detector ‘clicks’ and build statistics through
repetition.
However, in the context of solid-state systems, the details of the measurement process itself
are not fully understood and are an area of active research and recent progress. Single-shot
individual qubit measurements have been technically challenging, and the readout fidelity is
not yet as high as the fidelity of qubit operations (∼ 98 − 99% for single-qubit gates [108, 116]).
Each individual readout channel can provide an additional path for decoherence and must
also be calibrated. An example of the need for calibration is measurement cross-talk, which
can be significant in circuit-based architectures [99], but has now been suppressed to the 0.5%
level using an on-chip cavity as a filter [51]. Recently, the single-shot fidelity of independent
readouts of superconducting qubits has also been improved [51, 141] to ∼ 95%.
Alluded to previously, our circuit QED architecture provides access to an intriguingly
simple quadratic, or joint detector, where the measurement operator itself includes multi-
qubit correlations. In the last chapter (chapter 7) we observed the first steps towards using
the cavity as a joint qubit state detector. The coherence times in that experiment were
unfortunately too low for the generating high purity separable and entangled states. In this
chapter, we will show the full calibration and characterization of our joint detector and place
bounds of 2% on systematic deviations from the ideal joint measurement (section 8.4). This
is similar to determining the systematic errors, such as cross-talk [67], in individual readouts.
We then employ the joint detector for two qubits to perform quantum state tomography
for both separable states as well as highly entangled states, generated using the cavity bus
two transmon interaction (section 8.6). Furthermore, we demonstrate a high degree of
entanglement by measuring a large violation of a Clauser-Horne-Shimony-Holt inequality
[47] in a solid-state system, with a value of 2.61 ± 0.04, without optimizing for the target
state (section 8.7). Although not a strict test of local-hidden variable theories, our CHSH
 8.1. experimental setup 189

experiment serves as an entanglement witness and reflects the quality of both the separable
and entangled states generated in our system.

8.1 Experimental setup

The experiments presented in this chapter all involve the sample cQED222, reflecting an
improved design (chapter 5) of our standard multi-qubit circuit QED system. Although
still containing two transmon qubits, cQED222 also has on-chip independent transmon flux
tunability (section 5.3.3). This can be seen as an immediate improvement on the sample
cQED157, described in the cavity bus experiments presented in chapter 7.
The sample, as shown in figure 8.1, is a 4-port superconducting device comprising two
transmon qubits [61, 63] (which we will call QL , color-coded red, and QR , color-coded
blue) inside a microwave cavity bus, and flux-bias lines proximal to each qubit. The cavity,
normally off-resonance with the qubit transition frequencies fL and fR , couples the qubits
by virtual photon exchange and shields them from the electromagnetic continuum. As
previously discussed in chapter 7, microwave pulses resonant with fL or fR applied to the
cavity input port provide frequency-multiplexed single-qubit x- and y-rotations with high
fidelity [108] and selectivity [136]. Pulsed measurement of the homodyne voltage VH on the
cavity output port provides the joint qubit readout to be discussed later. The remaining two
ports create local magnetic fields that tune the qubit transition frequencies. Each qubit has
a split-pair of Josephson junctions, so its frequency is flux-tunable. By employing short-
circuited transmission lines with a bandwidth from dc to 2 GHz, we can tune fL and fR by
many GHz using room temperature voltages VL and VR (section 5.3.3).
Static tuning of qubit transitions using the flux-bias lines is demonstrated in figure 8.2.
This spectrum of single excitations shows the essential features of the cavity-coupled two-
qubit Hamiltonian and allows determination of relevant system parameters. Recall that the
Jaynes-Cummings Hamiltonian generalized (section 4.3.1) to multi-level transmon qubits is
N N
q q
H = ωC a† a + ∑ (∑ ω0 j ∣ j⟩q ⟨ j∣q + (a + a † ) ∑  jk ∣ j⟩q ⟨k∣q ). (8.1)
q∈{L,R} j=0 j,k=0

q
Here, ωC is the bare cavity frequency, ω0 j = ω0 j (ECq , EJq ) is the transition frequency for
q
qubit q from ground to excited state j, and  jk = q n jk (ECq , EJq ), with q a bare qubit-
cavity coupling and n jk a level-dependent coupling matrix element. These parameters will
depend on the qubit charging energy ECq and Josephson energy EJq . The flux control enters
190 entanglement and joint readout

fR VR
7 mm

fC
VH
fL fC

VL

300 µm
Figure 8.1: Schematic for two-qubit quantum bus with on-chip flux bias lines. Optical mi-
crograph of 4-port device with a coplanar waveguide cavity bus coupling two transmon qubits
(insets), and local flux-bias lines providing fast qubit tuning. Microwave pulses at the qubit
transition frequencies fL and fR drive single-qubit rotations, and a pulsed measurement of
the cavity homodyne voltage VH (at frequency fC ) provides two-qubit readout. The flux-bias
lines (bottom-left and top-right ports) are coplanar waveguides with short-circuit termination
next to their target qubit. The termination geometry allows current on the line to couple flux
through the split junctions.
 8.1. experimental setup 191

20 µm IR
01

10

cavity

I II III IV

Figure 8.2: Single excitation spectroscopy. Grey scale images of cavity transmission and
of qubit spectroscopy as a function of VR , showing local tuning of QR across the avoided
crossing with QL (point III) and across the vacuum Rabi splitting with the cavity (point IV).
Semi-transparent lines are theoretical best fits obtained from numerical diagonalization of a
generalized Jaynes–Cummings Hamiltonian. Preparation, single-qubit operations and mea-
surements are performed at point I, and a c-Phase gate for generating two-qubit entanglement
is achieved by pulsing into point II.
192 entanglement and joint readout

through EJq = EJq max

∣cos(πΦ q /Φ0 )∣, with Φ q the flux through the qubit loop, and a linear
flux-voltage relation Φ q = α qL VL + α qR VR + Φ q,0 , accounting for current crosstalk and offsets.
(Crosstalk, ∼ 30%, likely results from spatial distribution of flux-bias return currents on
the ground plane.) The above parameters are tightly constrained by the spectroscopy and
transmission data shown (figure 8.2) and other transmission data for the QL -cavity vacuum
Rabi splitting. Simultaneously fitting the spectra given by numerical diagonalization of
the Hamiltonian (truncated to N = 5 qubit levels and 5 cavity photons) to these data gives
max
EJL(R) /h = 28.48 (42.34) GHz, ECL(R) /h = 317 (297) MHz, L(R) /2π = 199 (183) MHz. Cavity
parameters are ωC /2π = 6.902 GHz and linewidth κ/2π = 1 MHz.
When the qubits are tuned to their maximum frequencies, point I in figure 8.2, they
are far detuned from the cavity and from each other, so that interactions are small. This
point is used for state preparation, single-qubit rotations and the joint measurement, in the
computational basis ∣0, 0⟩, ∣0, 1⟩, ∣1, 0⟩, and ∣1, 1⟩, where ∣l , r⟩ denotes excitation level l (r) for
QL (QR ). Operation at this point is also desirable because it is a flux sweet spot [63] for both
qubits, providing long coherence, with relaxation and dephasing times T1,L(R) = 1.3(0.79) µs
∗
and T2,L(R) = 1.8(1.15) µs, respectively. The coherence times are measured in the standard way
using a sliding π pulse protocol for the T1 determination and a Ramsey fringe interference
experiment for T2 .
Tuning QR into resonance with the cavity, point IV, reveals a vacuum Rabi splitting [52]
from which the qubit-cavity interaction strength is extracted. Tuning QR into resonance
with QL , point III, shows an avoided crossing resulting from a cavity-mediated, qubit-qubit
transverse interaction [91] as discussed in section 4.3.2 and experimentally investigated in
section 7.4. One might expect, given the emphasis on the swap oscillations from before, that
this would be the location for implementing a two-qubit entangling gate. However, we will
actually find that another interaction works better and will be the subject of the work in this
and chapter 9. Nonetheless, we can still characterize this transverse interaction region.

8.2 Virtual swap interaction via flux bias

The coupling strength between the two-qubits is found to be J/π = 105 MHz and can be seen
in the spectroscopy map at point III in figure 8.2. Whereas previously to turn on a qubit-qubit
swap, we were limited to using the ac-Stark interaction (section 7.4), here with the presence
of on-chip fast flux bias lines, it is possible to tune the swap interaction by simply modulating
the flux on one of the qubits.
 8.2. virtual swap interaction via flux bias 193

(a) (b)
M M
RLy (π) RLy (π) RLy (π) RRy (π)
(c) (d)
flux pulse M flux pulse M

RLy (π) RLy (π) RLy (π) RRy (π)

(a) (c) (b) (d)
250
∣, ⟩
(e)
200
Amplitude (µV)

∣, ⟩
150

100
∣, ⟩
50
∣, ⟩
0 20 40 60 80
Flux pulse duration (ns)

Figure 8.3: Flux bias swap experiments. (a)–(d) Protocols for flux-based swap oscillations. In
(a), the experiment is a sliding π-π on the left qubit, resulting in a finite population of ∣1, 0⟩
with increasing time between the pulses [shown in (e) dotted red trace]. In (b), the experiment
is a π on the left qubit followed by a variable delay and then π on the right qubit. The resulting
state ends up being near the ∣1, 1⟩ level [in (e) dashed blue trace]. Swap oscillations [solid red
and blue in (e)] can be seen when the flux pulse is turned on, for gate sequences (c) and for (d),
and out of phase from one another.

We can perform a protocol similar to that given in figure 7.9, except with the ac-Stark
pulse replaced by a flux pulse. Flux pulses are implemented on each of the lines (L and R)
using two channels of a Tektronix AWG 5014. The flux pulses are programmed to have a
rise time of 1 ns. The length of the flux pulse can be varied just as the Stark pulse length and
coherent swap oscillations can be seen, as shown in figure 8.3.
Due to the improved coherence times, these swaps can go on for much longer with higher
contrast. Furthermore, the slope in the swap oscillations which were observed with the Stark
swap are no longer present. We can also observe the anti-correlations between the oscillations
depending on applying the final π rotation on either the left qubit (figure 8.3c) or the right
qubit (figure 8.3d).
194 entanglement and joint readout

√
However, upon trying to use the flux swaps for performing a iSWAP gate in this system
proves to be difficult. The primary issue is that we cannot turn the swap interaction on fast
√
enough to perform the iSWAP. The ns-resolution of the flux pulses results is too short
compared to the swap frequency J/π, making an adiabatic state transfer more likely than the
swap. Fortunately, we can exploit the two-transmon σz ⊗ σz interaction described in chapter 4,
which does not require very fast tuning, but employ a slower, adiabatic flux control.

8.3 Higher-level transmon interaction

Looking back at the single excitation spectroscopy map of figure 8.2, the VR bias point at
which we we perform two-qubit interactions is at point II. At this voltage, which comes
slightly before the virtual qubit-qubit swap interaction, there are in fact no interactions which
are immediately apparent on examining the one-excitation manifold.
However, a useful two-qubit interaction is revealed in the two-excitation spectrum,
figure 8.4a. As VR is swept away from point I, the non-computational higher-level transmon
excitation ∣0, 2⟩ (left transmon in ground state, right transmon in second excited state),
decreases more rapidly than the computational state ∣1, 1⟩ (both transmons in their first
excited states). These two states in fact can be tuned into degeneracy at point II. However,
as shown in figure 8.4b, there is actually a large (160 MHz) cavity-mediated interaction
between these levels, resulting in a frequency shift ζ/2π of the lower branch with respect
to the sum fL + fR , in good agreement with a numerical diagonalization of the generalized
Jaynes–Cummings Hamiltonian. This avoided crossing causes the transition frequency to
∣1, 1⟩ to deviate from the sum of the transition frequencies to ∣0, 1⟩ and ∣1, 0⟩.
The two-excitation spectroscopy is performed using a pump and probe two-tone tech-
nique: one microwave excitation pulse is applied to the single-excitation transition frequen-
cies, f01 or f10 , before a second probe pulse is swept as a function of frequency, and the change
in homodyne transmission through the cavity is measured.
The shift of the transition frequency to ∣1, 1⟩ is the mechanism of an entangling conditional
phase (c-Phase) gate (section 2.3.2 and section 4.3.3). Recall that flux pulses, adiabatic with
 8.3. higher-level transmon interaction 195

02

11

20

I II III IV

01+10 02

11

20
II III

Figure 8.4: Two excitation spectroscopy. (a) Flux dependence of transition frequencies
from the ground state ∣0, 0⟩ to the two-excitation manifold. Two-tone spectroscopy mea-
surements (points) show an avoided crossing between the computational state ∣1, 1⟩ and the
non-computational state ∣0, 2⟩ at point II, in good agreement with numerical diagonalization
of the Hamiltonian (dashed curves). (b) Zoom-in on the avoided crossing, where we see that
the transition frequency to ∣1, 1⟩ deviates from the sum of the transition frequencies to ∣0, 1⟩
and ∣1, 0⟩.
196 entanglement and joint readout

respect to the ∣1, 1⟩ ↔ ∣0, 2⟩ avoided crossing, produce phase gates

⎛1 0 0 0 ⎞
⎜ ⎟
⎜0 e iθ z01 0 ⎟
⎜
U =⎜
0 ⎟
⎟ (8.2)
⎜0 0 e iθ 0 ⎟
10
z
⎜ ⎟
⎝0 0 0 e iθ z ⎠
11

in the computational Hilbert space. Here, θ zlr = 2π ∫ δf lr (t) dt is the dynamical phase
acquired by ∣l , r⟩, and δf lr is the deviation of f lr from its value at point I. A VR pulse into
point II such that ∫ ζ(t) dt = (2n + 1)π with integer n implements a c-Phase, because
θ z11 = θ z01 +θ z10 − ∫ ζ(t) dt. This method of realizing a c-Phase by adiabatically using the avoided
crossing between computational and non-computational states is generally applicable to qubit
implementations with finite anharmonicity, such as transmons [63] or phase qubits [116]. A
similar approach involving higher excitation levels with non-adiabatic pulses was previously
proposed [96]. The negative anharmonicity permits the phase gate at point II to occur before
the onset of transverse coupling at point III.
Control of ζ by two orders of magnitude provides an excellent on-off ratio for the c-Phase
gate. Measurements of ζ obtained from spectroscopy and from time-domain experiments
show very good agreement as shown in figure 8.5. The time-domain method measures the
difference in the precession frequency of QL in two Ramsey-style experiments where a VR -
pulse of varying duration (0–100 ns) is inserted between π/2 rotations of QL , with QR either
in the ground state ∣0⟩ or excited into state ∣1⟩. Using the time-domain approach, we measure
a residual ζ/2π ≈ 1.2 MHz at point I (indicated in figure 8.5 by the star). The theoretical ζ
obtained by numerical diagonalization shows reasonable agreement with the data, except for
a scale factor that is likely due to higher modes of the cavity, not included in the calculation.

8.3.1 Tuning up a c-Phase gate

We employ a protocol to tune up the c-Phase gates which is quite similar to a Ramsey fringe
experiment. As first alluded to in the previous section, the c-Phase is dependent upon ζ(t) as
well as the individual qubit dynamical phases θ z01 and θ z10 . Whereas the ζ(t) interaction needs
to be an adiabatic pulse with a large enough amplitude to enter the ∣0, 2⟩ ↔ ∣1, 1⟩ interaction
region, the individual qubit dynamical phases can be achieved with small flux excursions
which shift the qubit frequencies f01 and f10 via a z-rotation (section 8.6). Therefore, in our
 8.3. higher-level transmon interaction 197

II

Figure 8.5: Agreement of the splitting between experiment and theory. The coupling strength
ζ/2π = f01 + f10 − f11 of the effective σzL ⊗ σzR interaction, obtained both from spectroscopy
(solid curve) and from time-domain experiments (points) (see text for details). Numerical
diagonalization and perturbation theory for 3-level transmons agree reasonably with data.
The perturbation calculation diverges at the avoided crossing. Perturbation theory for 2-level
qubits gives the wrong magnitude and sign for ζ, and demonstrates that the higher transmon
excitations are necessary for the interaction. Time-domain measurement and theory both
give ζ/2π ≃ 1.2 MHz at point I. The tunability of ζ over two orders of magnitude provides an
excellent on-off ratio for the two-qubit c-Phase gate.

implementation, to tune up a c-Phase gate, we have to combine a strong flux pulse on the
right qubit FBL with weaker flux pulses on both the right and left FBLs.
Since the qubits start from being at point I for the single-qubit operations, the flux bias
lines are both originally set to nominal DC voltages which tune to this location, which we
can call VL0 and VR0 . The four different c-Phase gates differ by whether θ z01 and θ z10 are even or
odd multiples of π: we fine tune θ z01 with small adjustments to the rising and falling edges
of the VR -pulse, and θ z10 with the amplitude of a simultaneous weak VL -pulse as shown in
figure 8.6. The pulse onto the right flux bias line is essentially a bi-level pulse, with the larger
amplitude ∆VRζ sufficient to tune into the conditional phase interaction ζ(t) and the lower
amplitude ∆VR0 at the start and end to tune the dynamical z-phase on the right qubit. The left
198 entanglement and joint readout

∆VRζ ∆VRζ

(a) ∆VL
(c) ∆VR
∆VL
M M
RLy (π/) RLy (π/) RLy (π/) RLy (π/)
∆VRζ ∆VRζ

(b) (d) ∆VR

∆VL ∆VL
M M

RLy (π/) R Ry (π/) RLy (π/) R Ry (π/)

350
(e) (f )
300
Amplitude (mV)

250
200
150
100 (a) (c)
(b) (d)
50
0 20 40 60 80 100 120 0 20 40 60 80 100 120
Left Flux Pulse Amplitude (arb) Right Flux Ledge Amplitude (arb)

Figure 8.6: Conditional phase gate tune-up sequences. Two sets of experiments are used to
tune up the flux pulses for the conditional phase gates. (a) and (b) involve first applying a π/2
pulse on the left qubit, followed by a fixed flux pulse of ∆VRζ on the right FBL combined with a
left FBL pulse with a varying amplitude ∆VL0 , and then a final π/2 pulse on either the left or
right qubit. The second set (c) and (d) uses a similar pulse sequence, but with varying the ledge
amplitude, ∆VR0 , on the rising and falling edges of the pulse on the right FBL. The measured
homodyne responses are shown in (e) and (f). In (e), the ∆VL0 is set such that the solid red and
blue curves are exactly out of phase, here at ∼ 79. Then, in (f), depending on tuning ∆VR0 to
∼ 20 or ∼ 70 will define the c-Phase gate corresponding to cU01 or cU11 .

qubit pulse is simply a square pulse with a single amplitude level ∆VL0 used to tune the left
qubit dynamical phase.
We employ two experiments in a two-step procedure for tuning the flux bias line levels
for the c-Phase gates. The first experiment is built up of two sequences: in the first we apply a
R y (π/2) pulse to the left qubit, turn on a large right FBL pulse ∆VRζ to get into the c-Phase
interaction region, while also turning on a left FBL pulse of varying amplitude ∆VL0 , followed
by a final pulse on the left qubit which is again a R y (π/2); the second sequence is identical
to the first except at the last stage we also add an additional R y (π) to the right qubit. We
expect to see oscillations in each case, and look for the appropriate ∆VRζ and ∆VL0 such that
 8.4. joint readout of two qubits 199

the oscillations in the two traces are out-of-phase. In the second sequence, the different state
of the right qubit affects the phase of the oscillations. Therefore, we are probing the amount
of phase necessary on the left qubit in order to realize the conditional-phase flip. The protocol
and a sample experimental output is shown in figure 8.6a.
The second step is again built up of two sequences, and aims to unwrap the dynamical
phase on the right qubit: for the first sequence, (1) apply R y (π/2) on to the right qubit; (2)
turn on the large right FBL pulse of amplitude ∆VRζ and left FBL pulse of amplitude ∆VL0
(both determined from the first step) while sweeping the offset voltage on the right FBL ∆VR0 ;
(3) apply R y (π/2) on to the right qubit; for the second sequence, same as the first except
steps (1) and (3) apply R y (π) onto the left qubit as well. Again, we expect to see oscillations
depending on the amplitude of the right FBL offset voltage. Depending on the selection of
∆VR0 will allow us to tune-up any of the four cU i j conditional phase gates (see figure 8.6b).

8.3.2 Generating Bell states

With access to the four different c-Phase gates (cU i j ) as well as single-qubit rotations, it is
then possible to generate maximally entangled states, such as the four Bell states,
1 1
∣Ψ± ⟩ = √ (∣0, 0⟩ ± ∣1, 1⟩) ∣Φ± ⟩ = √ (∣0, 1⟩ ± ∣1, 0⟩) . (8.3)
2 2
The pulse protocols for their generation are summarized in figure 8.7. In each case, the scheme
involves first to place both qubits in a superposition state using individual single-qubit π/2
rotations. That is followed by the application of any of the 4 different c-Phase gates. At this
point the two qubits are in fact already in an entangled state. To make this more evident, a
single-qubit rotation of π/2 on one of the qubits will result in one of the four Bell states.
Simply the ability to generate entangled states is not enough however, as it is imperative
to be able to readout the state and figure out exactly how accurately are the experimentally
created states from what is theoretically expected. Therefore, we next build upon the ideas of
using the cavity as a joint readout and state tomography discussed in section 4.4.2 and apply
it to both separable and entangled states.

8.4 Joint readout of two qubits

To accurately and precisely detect two-qubit states with our cavity bus, we first seek a complete
physical model and calibration of the joint readout. The physical mechanism enabling the
200 entanglement and joint readout

0 R y ( π ) R y ( π ) 0 R y ( π ) R y ( π )
+
cU ∣Ψ ⟩ cU ∣Ψ− ⟩
0 R y ( π ) 0 R y ( π )

0 R y ( π ) R y ( π ) 0 R y ( π ) R y ( π )
cU ∣Φ+ ⟩ cU ∣Φ− ⟩
0 R y ( π ) 0 R y ( π )

Figure 8.7: Experimental protocols for generating Bell states. Gate sequence generating two-
qubit entanglement. Starting from ∣0, 0⟩, simultaneous π/2 rotations on both qubits create
an equal superposition of the four computational states. A c-Phase cU i j then phase shifts
∣i, j⟩ in the superposition and produces entanglement. A final π/2 rotation on QL evolves the
entangled state into one of the four Bell states depending on the cU i j applied.

joint readout is a qubit-state-dependent dispersive cavity shift that is large relative to the
cavity linewidth κ/2π = 1 MHz. In this ‘strong dispersive’ regime [110], recall that the system
is described by a dispersive Jaynes-Cummings Hamiltonian

ωL ωR
HJC /ħ = (ωC + χL ZI + χR IZ)a † a − ZI − IZ, (8.4)
2 2
where ωC is the bare resonator frequency, ωL(R) is the first excited state transition frequency
for the left (right) qubit, and χL(R) is the left (right) qubit-state dependent cavity shift. Note
that we have assumed that the qubits are also far enough detuned from one another that we
drop the virtual-swap interaction term. The cavity shifts χL,R are determined by a pulsed
measurement of the transmitted homodyne voltage VH , having prepared each of the four
computational basis states (∣0, 0⟩, ∣0, 1⟩, ∣1, 0⟩, ∣1, 1⟩) using single-qubit gates. Figure 8.8a–d
show the transient in ⟨VH ⟩ as a function of drive frequency ωRF (ensemble average of 600,000
repetitions). On time scales shorter than the qubit relaxation times, t ≲ T1 = 1.2(0.9) µs,
L(R)

the largest transmission occurs at distinct frequencies (Fig. 1f shows a time-averaged voltage
∆t
V̄H = ⟨∫0 VH dt⟩/∆t with ∆t = 0.5 µs), from which we estimate χL(R) /2π = 13(4) MHz.
When the shifts are many linewidths, χL , χR ≫ κ, and qubit relaxation during measure-
ment is negligible, ∆t ≪ T1 , driving with a tone at the cavity frequency corresponding to ∣0, 0⟩
would query the joint property that both qubits are in the ground state: transmission is high
when the state is projected onto ∣0, 0⟩ and zero otherwise. In this ideal scenario, V̄H = Tr(ρM),
where ρ is the two-qubit density matrix and M ∝ ∣0, 0⟩ ⟨0, 0∣ = (I + ZI + IZ + ZZ)/4 is the
measurement operator. However, qubit relaxation during the measurement and partial over-
 8.5. calibrating the measurement model 201

3 a |0,0 e
2 |1,1 |1,0 |0,1 |0,0
1.0

(mV)
1
0
3 b 0.5
|0,1
2
Time (ms)

1 0.0
0 6.78 6.80 6.82
ωRF/2p (GHz)
3 c |1,0
2 f
1 1.5

(mV)
0
1.0
3 d |1,1
2 0.5
1
0 0.0
6.78 6.80 6.82 0 1 2 3 4
ωRF/2p (GHz) Time (µs)

Figure 8.8: Measurement transients for joint readout. Transmitted VH amplitude as a func-
tion of time and cavity drive frequency ωRF , for computational basis states (a) ∣0, 0⟩, (b) ∣0, 1⟩,
(c) ∣1, 0⟩, and (d) ∣1, 1⟩. (e) Time average of the VH transients in (a–d) over the first 500 ns.
(f) Transients of VH for ωRF at the cavity frequency corresponding to ∣0, 0⟩.

lap of the dispersive peaks, evident in figure 8.8e, make the measurement operator take the
more general form

M = β II II + β ZI ZI + β IZ IZ + β ZZ ZZ, (8.5)

where β LR are real calibration constants that must be determined.

8.5 Calibrating the measurement model

A comprehensive test of this measurement model is performed with a sequence of Rabi-

flopping experiments. Figure 8.9a–b show V̄H as a function of the duration of an applied drive
at ωL and ωR , respectively. In each case the drive induces a Rabi oscillation of the addressed
qubit around the y axis of its Bloch sphere. A third experiment figure 8.9c measures V̄H in
response to simultaneous driving of both qubits. The observed oscillations show frequency
components at the individual qubit Rabi frequencies, ΩL , ΩR , but also at the sum and
202 entanglement and joint readout

Rabi expt. left right both

Model
a
1.5

0.5
b
(mV)

1.5

0.5

1.5 c

0.5
0.0 0.1 0.2 0.3 0.4
Rabi drive length (ms)
Figure 8.9: Rabi experiments for readout characterization. Rabi oscillations on the (a) left
qubit, (b) right qubit, and (c) simultaneously on both. Solid lines are fits to the model in (8.5).

difference due to the ZZ term in Eq. (2). This is clearly revealed in the Fourier transform of
the oscillations shown in figure 8.10.
We these oscillations with the most general two-qubit measurement operator

M= ∑ β LR LR, (8.6)
L,R∈{I,X,Y ,Z}

using theoretical expressions for ⟨Z⟩ and ⟨X⟩ assuming independently driven qubits. Because
in these tests each qubit is driven around the y axis of its Bloch sphere, all terms involving
Y L and Y R in (8.6) would not contribute to V̄H . The presence of such terms can be tested
by rotating each or both qubits around their x axis instead. We do not find any significant
differences in such experiments from the ones presented in the text, and the results here can
be generalized for both quadratures X and Y.
In our experiment the detuning ∼ 1.5 GHz between the two qubits is large compared to
the Rabi-flopping rates, and we can assume a simple model of independent qubit driving.
For a qubit driven at a rate Ω around its y axis starting from the ground state, the theoretical
 8.5. calibrating the measurement model 203

Rabi expt. left right both

Model

100 FT of expt.
FT amplitude (mV)

FT of model

10

20 40 60
Frequency (MHz)
Figure 8.10: Fourier transforms of Rabi oscillations. Fourier transform (FT) of the three Rabi
experiments (circles) and of best fits (lines). While the red (blue) traces show one main peak at
the Rabi frequency ΩL(R) , the purple traces reveal peaks at ΩL , ΩR , ΩL + ΩR , and ΩL − ΩR ,
demonstrating the mixing property that makes the joint measurement sensitive to qubit-qubit
correlations.

time evolution of ⟨Z⟩ and ⟨X⟩ is given by

γ1 γ2 e −t/τ R Ω2 sin(Ω̃t)
⟨Z⟩(t) = + (cos(Ω̃t) + ), (8.7a)
γ1 γ2 + Ω 2 γ1 γ2 + Ω 2
τ R Ω̃
γ1 Ω e −t/τ R Ω ⎛ [2Ω2 + γ1 (γ2 − γ1 )] sin(Ω̃t) ⎞
⟨X⟩(t) = − γ 1 cos( Ω̃t) − . (8.7b)
γ1 γ2 + Ω2 γ1 γ2 + Ω2 ⎝ 2Ω̃ ⎠
√
Here, Ω̃ = Ω2 − (1/τ R )2 is an effective oscillation rate, γ1 = 1/T1 is the relaxation rate,
γ2 = γ1 /2 + γ ϕ is the dephasing rate, and τ R = 2/(γ1 + γ2 ) is the Rabi decay time.
204 entanglement and joint readout

The best fits to the oscillations place bounds on deviations from the measurement model
of (8.5). Using these expressions in the full model, (8.6), and fitting to the three experiments,
we estimate the coefficients β LR . For single-qubit driving (figure 8.9a–b), the right (left) qubit
is always in the ground state, and only terms ⟨ZI⟩, ⟨XI⟩, ⟨XZ⟩ and ⟨ZZ⟩ (⟨IZ⟩, ⟨IX⟩, ⟨Z X⟩,
and ⟨ZZ⟩) contribute to the V̄H oscillation. Using the form

V̄Ha = W0 + W1 ⟨ZI⟩ + W2 ⟨XI⟩, (8.8a)

V̄Hb = W0 + W1 ⟨IZ⟩ + W2 ⟨IX⟩, (8.8b)

with W0 , W1 , W2 , ΩL(R) , γ1L(R) , and γ2L(R) as free parameters gives an excellent fit. In both cases,
the best-fit W2 , corresponding to β XI(IX) + β XZ(ZX) , is less than 2% of the full range of V̄H ,
∼ 2β IZ + 2β ZI . For the doubly-driven case (figure 8.9c), the fit function used is

V̄Hc =β II + β XI ⟨XI⟩ + β ZI ⟨ZI⟩ + β IX ⟨IX⟩ + β IZ ⟨IZ⟩

+ β XX ⟨XX⟩ + β XZ ⟨XZ⟩ + β ZX ⟨Z X⟩ + β ZZ ⟨ZZ⟩, (8.9a)

with β i j , ΩL , ΩR , γLj , and γRj as fit parameters. The best-fit coefficients captured in (8.5) are
(β II , β IZ , β ZI , β ZZ ) = (800, 380, 380, 200) µV. Best-fit values of the remaining coefficients
are each less than 2% of the full range of V̄H .
These Rabi experiments thus corroborate (8.5) and give the calibration (β II , β IZ , β ZI , β ZZ )
= (800, 380, 380, 200) µV . The jointness, defined as β ZZ /β IZ(ZI) , is 0.6, indicating the high
sensitivity of the readout to qubit-qubit correlations. This high relative sensitivity to two-qubit
correlations in the measurement operator, or jointness, makes the joint readout as efficient
for measuring qubit correlations as for single-qubit polarizations. Since the correlation is
performed before averaging, the classical amplifier noise that limits the single-shot readout
fidelity enters only as a statistical error, and can be largely eliminated with sufficient repetition.
Therefore, this fully-characterized joint readout will be sufficient to perform full two-qubit
state tomography.

8.6 Quantum state tomography and the Pauli set

Having characterized the joint readout, we can now perform quantum state tomography of
separable and entangled two-qubit states generated using the C-phase gate. We extend ref. 4,
where two-qubit state tomography with a joint readout was first demonstrated, by obtaining
an overcomplete set of 30 measurements through applying different pairs of simultaneous
 8.6. quantum state tomography and the pauli set 205

single-qubit rotations prior to detection as shown in table 8.1. These measurements involve
applying different simultaneous rotations on the qubits. The 15 measurements labeled M i
involve positive rotations chosen from {I, R +π , R y }. The remaining 15, labeled N i ,
+π/2 +π/2
x , Rx
involve negative rotations chosen from {I, R−π , R y }. Ensemble averages of M i and
−π/2 −π/2
x , Rx
N i are obtained by repeating state preparation, analysis rotation, and measurement 600,000
times. Although just 15 linearly independent measurements (such as either all M i or all N i )
is sufficient for state tomography, using all of these rotations and least-squares estimation
reduces the statistical and systematic error in the final extraction of either the density matrix
ρ or the Pauli set P⃗ discussed in the next two sections.

8.6.1 The density matrix representation

The set of measurements described above can be used to estimate the density matrix ρ for
describing the quantum mechanical state using the prescription described in section 2.5.2.
Recall that for any two-qubit quantum state we can choose a set of 16 linearly independent
operators {M i } such that ρ can be decomposed as
16
ρ = ∑ ci Mi ,
i=1

where the set {c i } are the 16 parameters to be estimated. If the operators are observables,
then the 16 expectation values m i = Tr[M i ρ] determine c j by
16
m i = ∑ Tr[M i M j ]c j .
j=1

Only 15 independent (either all the positive, or all the negative) measurements are needed
to determine ρ because of the constraint of trace normalization, tr ρ = 1 (equivalently we
choose M16 = I, which always gives m16 = 1). While ideally ρ could be obtained from the
experimental m i by inversion of Tr[M i M j ], this method pays no attention to the properties
ρ must have: Hermiticity and positive semi-definiteness (trace normalization is included by
the choice of decomposition). However, by following the Maximum Likelihood Estimation
(MLE) technique from section 2.5.2, it is possible to obtain estimates to the density matrices.
The inferred density matrices ρml for the four Bell states are shown in figure 8.11. We
characterize the quality of these states through metrics which are computable from ρml : purity
(section 2.6) given by P(ρ) = tr(ρ2 ), fidelity to the target state ∣ψ⟩ given by F(ρ, ψ) = ⟨ψ∣ρ∣ψ⟩,
206 entanglement and joint readout

∣Ψ+ ⟩ = √ (∣⟩ + ∣⟩) ∣Ψ− ⟩ = √ (∣⟩ − ∣⟩)

 

a b

0.5

0
-0.5 11
10
00 01 01
10 00
11
c d

∣Φ+ ⟩ = √ (∣⟩ + ∣⟩) ∣Φ− ⟩ = √ (∣⟩ − ∣⟩)

 

Figure 8.11: Density matrix representation of Bell states. Real part of maximum-likelihood
density matrix ρml of the entangler output for cU10 , cU00 , cU11 , and cU01 , respectively (imag-
inary elements of ρml are less than 0.03, 0.02, 0.07, 0.08). Extracted metrics for the four
entangler outputs include purity P = 0.87 ± 0.02, 0.92 ± 0.02, 0.88 ± 0.02, 0.79 ± 0.03, fidelity
to the ideal Bell state F = 0.91 ± 0.01, 0.94 ± 0.01, 0.90 ± 0.01, 0.87 ± 0.02 and concurrence
C = 0.88 ± 0.02, 0.94 ± 0.01, 0.86 ± 0.02, 0.81 ± 0.04. The uncertainties correspond to the
standard deviation in 16 repetitions of generation-tomography for each entangler.
 8.6. quantum state tomography and the pauli set 207

Table 8.1: The 30 raw measurements.

Pre-rotation Measurement operator

M01 I⊗I +βZI ZI + βIZ IZ + βZZ ZZ
π
M02 Rx ⊗I −βZI ZI + βIZ IZ − βZZ ZZ
π
M03 I⊗Rx +βZI ZI − βIZ IZ − βZZ ZZ
π/2
M04 Rx ⊗I +βZI Y I + βIZ IZ + βZZ Y Z
π/2 π/2
M05 Rx ⊗Rx +βZI Y I + βIZ IY + βZZ Y Y
π/2 π/2
M06 Rx ⊗Ry +βZI Y I − βIZ IX − βZZ Y X
π/2 π
M07 Rx ⊗Rx +βZI Y I − βIZ IZ − βZZ Y Z
π/2
M08 Ry ⊗I −βZI XI + βIZ IZ − βZZ XZ
π/2 π/2
M09 Ry ⊗Rx −βZI XI + βIZ IY − βZZ XY
π/2 π/2
M10 Ry ⊗Ry −βZI XI − βIZ IX + βZZ XX
π/2 π
M11 Ry ⊗Rx −βZI XI − βIZ IZ + βZZ XZ
π/2
M12 I⊗Rx +βZI ZI + βIZ IY + βZZ ZY
π π/2
M13 Rx ⊗Rx −βZI ZI + βIZ IY − βZZ ZY
π/2
M14 I⊗Ry +βZI ZI − βIZ IX − βZZ ZX
π π/2
M15 Rx ⊗Ry −βZI ZI − βIZ IX + βZZ ZX
N01 I⊗I +βZI ZI + βIZ IZ + βZZ ZZ
−π
N02 Rx ⊗I −βZI ZI + βIZ IZ − βZZ ZZ
−π
N03 I⊗Rx +βZI ZI − βIZ IZ − βZZ ZZ
−π/2
N04 Rx ⊗I −βZI Y I + βIZ IZ − βZZ Y Z
−π/2 −π/2
N05 Rx ⊗Rx −βZI Y I − βIZ IY + βZZ Y Y
−π/2 −π/2
N06 Rx ⊗Ry −βZI Y I + βIZ IX − βZZ Y X
−π/2 −π
N07 Rx ⊗Rx −βZI Y I − βIZ IZ + βZZ Y Z
−π/2
N08 Ry ⊗I +βZI XI + βIZ IZ + βZZ XZ
−π/2 −π/2
N09 Ry ⊗Rx +βZI XI − βIZ IY − βZZ XY
−π/2 −π/2
N10 Ry ⊗Ry +βZI XI + βIZ IX + βZZ XX
−π/2 −π
N11 Ry ⊗Rx +βZI XI − βIZ IZ − βZZ XZ
−π/2
N12 I⊗Rx +βZI ZI − βIZ IY − βZZ ZY
−π −π/2
N13 Rx ⊗Rx −βZI ZI − βIZ IY + βZZ ZY
−π/2
N14 I⊗Ry +βZI ZI + βIZ IX + βZZ ZX
−π −π/2
N15 Rx ⊗Ry −βZI ZI + βIZ IX − βZZ ZX

and concurrence (section 2.6.1) C. Note that there are several common definitions of fidelity
in the literature. Our definition is the square of the fidelity used in [67] and [142]. The
extracted metrics for the four separate Bell state cases are P = 0.87 ± 0.02, 0.92 ± 0.02, 0.88 ±
0.02, 0.79±0.03, fidelity to the ideal Bell state F = 0.91±0.01, 0.94±0.01, 0.90±0.01, 0.87±0.02
and concurrence C = 0.88 ± 0.02, 0.94 ± 0.01, 0.86 ± 0.02, 0.81 ± 0.04. The uncertainties
correspond to the standard deviation in 16 repetitions of generation and state tomography
208 entanglement and joint readout

for each sequences. These values significantly extend the state of the art for solid-state
entanglement [67], and provide evidence that we have a high-fidelity universal set of two-
qubit gates.

8.6.2 Biasing of metrics by maximum-likelihood estimation

Although the density matrix representation is the more common method of depicting quan-
tum state tomography, the maximum-likelihood estimation often associated with the density
matrix determination can become biased if the true mean lies close to a boundary of the
allowed parameter space [143]. Specifically, as the fidelity of the states become closer and
closer to ideal, a biased estimation via maximum-likelihood becomes more probable.
In order to quantify the importance of this effect on the estimation of lower bounds
on concurrence C given by entanglement witnesses (discussed in the next section), we can
perform Monte-Carlo simulations for nearly-pure Werner states [39],

ρW (λ) = λ ∣ψ − ⟩ ⟨ψ − ∣ + (1 − λ)I/4, (8.10)

with Werner parameter λ ∈ [0.8, 1]. We can create 100 sets of simulated raw measurements
for each λ by assuming Gaussian amplifier noise consistent with our experiments. Figure 8.12
shows a lower bound on the concurrence Cbound as a function of the true C of the Werner
state, obtained with and without MLE processing of the simulated noisy data. We find that
while the mean of Cbound estimated directly from the raw data is unbiased, the mean of the
concurrence bound obtained with MLE becomes increasingly biased the more pure the
Werner state, i.e., the closer λ is to unity. MLE underestimates the bound by 1% at C = 0.85,
and by 4% at C = 1.
Therefore, although using MLE for obtaining a density matrix can be a very useful
technique to get at a real physical state, it can also be a highly non-linear process which can
result in the incorrect estimation of certain metrics associated with the states themselves.
Furthermore, while errors on the individual measurements are Gaussian, the propagation of
such errors through the estimation process of the density matrix is not very straightforward.
The best technique is to Monte Carlo simulate the noise on the extraction of the density matrix
given the raw experimental measurements and their associated Gaussian noise variance.
However, we can attempt to process and visualize the measurements in an alternative way
that will allow linear propagation of errors while avoiding MLE.
 8.6. quantum state tomography and the pauli set 209

1.1 B ounds , without MLE

B ounds , with MLE
Average bound, without MLE
1.0 Average bound, with MLE
T rue bound
Concurrence bound

0.9

0.8

0.7

0.6

0.70 0.75 0.80 0.85 0.90 0.95 1.00

Concurrence,

Figure 8.12: Bias of entanglement metrics from MLE. Comparison of the lower bound Cbound
on concurrence C computed from simulated noisy raw data with and without use of MLE.
The bound computed using MLE systematically underestimates the true bound (in this case,
always equal to the true concurrence C, red line), while the bound computed directly from the
simulated raw data remains faithful even as the Werner state approaches the Bell state ∣Ψ− ⟩,
i.e., as λ → 1. For ∣Ψ− ⟩, the MLE-computed bound underestimates the true bound by 4%.

8.6.3 The Pauli set representation

The Pauli set P⃗ is another representation of the full quantum state which can be a handy
tool for visualization, systematic errors determination, and statistical error propagation. A
linear least-squares estimator can be used to extract the Pauli set P, ⃗ whose 16 elements are
the expectation values of the two-qubit Pauli operators, ⟨LR⟩, where L, R ∈ {I, X, Y , Z}. The
Pauli set is related to the raw measurements as given in table 8.1. The two-qubit density matrix
(without MLE) is linearly related to the elements of P⃗ by ρ = ∑L,R LR⟨LR⟩/4.
An advantage of examining the set P⃗ is that it is a simple visual tool to distinguish separable
from entangled states. In figure 8.13, we show two separable (a-b) and two entangled (c-d)
states. For pure states, P⃗ ideally contains three non-zero bars, all of unit magnitude. P⃗ can
210 entanglement and joint readout

be divided into three sections: the single-qubit polarization vectors, P⃗L = {XI, Y I, ZI} (red)
and P⃗R = {IX, IY , IZ} (blue), and the vector of two-qubit correlations Q ⃗ = {⟨XY⟩, ⋯, ⟨ZZ⟩}
(purple). For the separable states, we observe near unity components in the three sections.
In contrast, for the entangled states, only Q ⃗ has components with near-unity magnitude.
The presence of strong correlations and vanishing qubit polarization is a signature of a high
degree of entanglement.
To test for systematic errors we measure P⃗ for a collection of states that differ only by
a single-qubit rotation prior to measurement. These errors in detection could appear as
offsets or amplitudes that exceed the ±1 range of the elements of P. ⃗ Two such experiments
involving a rotation θ of the left qubit about its y axis having prepared the separable state
∣0, 0⟩ (experiment I) and the entangled state (∣0, 0⟩+∣0, 1⟩−∣1, 0⟩+∣1, 1⟩)/2 (experiment II) are
shown in figure 8.14a–b, respectively. In experiment I, ⟨XI⟩, ⟨ZI⟩, ⟨XZ⟩, and ⟨ZZ⟩ oscillate
with an average visibility of 97.6 ± 0.3%. Moreover, the measured amplitude of all the ideally-
zero bars is less than 0.1. In experiment II, the dominant oscillating components are all in
⃗ indicating that the state remains entangled throughout. In this case, we find a visibility
Q,
of 91.5 ± 0.3%, in good agreement with a master equation simulation incorporating qubit
relaxation and dephasing. An oscillation amplitude of ∼ 0.1 is observed in ⟨XI⟩ and ⟨ZI⟩, a
factor ∼ 2 larger than expected from theory. This discrepancy can arise from a combination of
small calibration errors in single-qubit rotations and various residual higher order couplings.
For example, the discrepancies between the experiment and master equation simulation
can arise from a systematic under-rotation of both qubits by only 1%. There are also higher
order couplings that are relevant at this level. The first is the finite strength of the two-
qubit ZZ entangling interaction [126] even in the off state (ζ/2π ∼ 1.2 MHz ). This residual
coupling leads to errors in some of the two-qubit correlations on the order of ζ/ΩL(R) ∼ 2%. A
second is the presence of a residual qubit-qubit interaction [136], as discussed in section 4.3.2,
(J/2π ∼ 60 MHz), that can lead to errors of order J/(ωL − ωR ) ∼ 4%. Another effect is
the qubit-state dependent filtering of the drive applied to a qubit, which is expected to be
on the order of χR(L) /(ωL(R) − ωC ) ∼ 2%. The effect of these couplings can be mitigated by
implementing appropriate composite pulse schemes [144] and will be explored in the future.
 8.6. quantum state tomography and the pauli set 

XI YI ZI IX IY IZ XY XZ YX YZ ZX ZY XX YY ZZ
1
0.5
0
0 -0.5
00
a 01
-1 10
11 00 01 10 11
1

0
Element value

-1 b

c
-1

-1 d
XI YI ZI IX IY IZ XY XZ YX YZ ZX ZY XX YY ZZ
Pauli measurement
Figure 8.13: Pauli set representation of two-qubit states. Experimental Pauli set (with trivial
⟨II⟩ =  not shown), for separable states a ∣, ⟩ and b (∣, ⟩ − ∣, ⟩ + ∣, ⟩ − ∣, ⟩)/ and
entangled states c ∣Ψ+ ⟩ and d (∣, ⟩ + ∣, ⟩ − ∣, ⟩ + ∣, ⟩)/. Red (blue) bars correspond to left
(right) single-qubit Pauli operators. Purple bars are the qubit-qubit correlations. The fidelities
to the four targeted states are F = . ± .%, . ± .%, . ± .%, and . ± .%. The
real part of the density matrix obtained using constrained maximum-likelihood estimation on
the same raw measurements is shown in the three-dimensional plot to the right of each Pauli
set.
212 entanglement and joint readout

a
180
0
0
0
-180

(deg) 180
b
0
C- 0
phase
0
-180
XI YI ZI IX IY IZ XY XZ YX YZ ZX ZY XX YY ZZ
Pauli measurement
Figure 8.14: Pauli set for separable and entangled states differing only by a single-qubit
rotation. Gate sequences and measured P, ⃗ having subjected a (experiment I) the separable
state ∣0, 0⟩ and b (experiment II) the entangled state (∣0, 0⟩ + ∣0, 1⟩ − ∣1, 0⟩ + ∣1, 1⟩)/2 to a rotation
R y (θ) on the left qubit, −210○ ≤ θ ≤ 210○ . In experiment I, the left qubit polarization rotates
along the x-z plane, while the right qubit remains fully-polarized along z. In experiment
II, both qubit polarizations vanish, with the only nonzero and oscillating Pauli operators
being qubit-qubit correlators (purple bars). Arrows at θ = −90(+90)○ indicate when the ideal
two-qubit state is the Bell state ∣Φ− ⟩ (∣Ψ+ ⟩).

8.7 Characterizing the quantum states

Although we have mentioned a few metrics which characterize the state and entanglement
using the full density matrix, we can now work with another set of metrics that use the Pauli
set of measurements, and permit the linear propagation of statistical measurement errors.

8.7.1 Fidelity to targeted states

In section 2.5.2, we introduced the fidelity (2.42), F = ⟨ψ∣ ρ ∣ψ⟩ to a targeted state ∣ψ⟩ quantifies
the control over two-qubit states, and is given by

F = P⃗ ⋅ P⃗target .
1
(8.11)
4
For experiment I and II, we find F = 98.8±1.0% and 93.4±1.5% (averaged over θ), respectively.
We find excellent agreement between experiment (black circles) and simulation (solid),
demonstrating the accuracy of both the state preparation and the measurement (Figs. 4c–d).
 8.7. characterizing the quantum states 213

8.7.2 Entanglement witnesses

Measures beyond fidelity are necessary to quantify the degree of two-qubit entanglement.
Often, entanglement monotones such as concurrence [39] C are obtained using non-linear
estimators. As previously mentioned, it is standard practice to first perform maximum-
likelihood estimation [145] to generate a positive ρ despite statistical or systematic errors
in the measurements, and to then calculate these metrics from the eigenvalue spectrum of
related matrices [39]. This non-linear process complicates the propagation of any statistical
and systematic errors in the measurements. It can also bias the estimation of metrics such as
C when the purity of the two-qubit state is high compared to the readout fidelity[146, 147], as
is typically the case with superconducting qubits.
Although not entanglement monotones, entanglement witnesses [39, 45] do allow us to
make quantitative statements about entanglement while using only linear operations on mea-
surements. To review, an entanglement witness W is a unity-trace observable with a positive
expectation value for all separable states, such that tr(ρW) < 0 guarantees entanglement.
Furthermore, B = −2 tr(ρW) gives a lower bound [45] on C. The optimal witnesses (strictest
lower bound) for the Bell states are
1
WΨ± = (II ∓ XX ± Y Y − ZZ), (8.12a)
4
1
WΦ± = (II ∓ XX ∓ Y Y + ZZ). (8.12b)
4
We return to the experiments described above I and II of the measured Pauli sets for
the separable and entangled states. The measured bounds on concurrence for experiment I
(II) are shown in figure 8.15 (figure 8.16). In experiment I, the four bounds are non-positive
for all θ to within measurement error, indicating that entanglement is not witnessed. This is
expected, since single-qubit rotations should not produce any entanglement. In figure 8.16
in contrast, bounds BΨ+ and BΦ− extend into the positive region, reaching 85.9 ± 1.5% and
88.1 ± 1.5% at θ = −90○ and 90○ , respectively. There is at least one positive bound for most
θ (excluding ±180○ and 0○ ), indicating that the two qubits are entangled. Agreement with
the master equation simulation (solid lines) in figure 8.15 and figure 8.16 shows the accuracy
of the entanglement witnesses and the small residuals ∼ 2% demonstrate the precision of
measurement by joint readout.
214 entanglement and joint readout

fidelity bounds
data
theory

-1
-180 -90 0 90 180
Rotation angle θ (deg)

Figure 8.15: Entanglement witness for separable states. Experimental lower bounds Bi
(orange) on concurrence given by the optimal witnesses for Bell states WΨ+ (circles),
WΨ− (squares), WΦ+ (triangles), and WΦ− (crosses), and fidelity F to the ideal state (black
circles) for experiment I. Solid curves are results of master equation simulations.

8.7.3 Clauser-Horne-Shimony-Holt inequality violation

A related entanglement measure is the Clauser-Horne-Shimony-Holt (CHSH) operator, often
used to test quantum mechanics against local-hidden variable (LHV) theories. A CHSH
operator [47] is defined as

CL,R,L′ ,R′ = LR + LR′ + L′ R − L′ R ′ , (8.13)

with {L, L′ } and {R, R ′ } being pairs of single-qubit Pauli operators along any two axes of the
left and right qubits, respectively. With a general choice of axes, for separable states, ∣⟨C⟩∣ ≤ 2,
coinciding with the LHV bound. For the specific choice L ⊥ L′ and R ⊥ R ′ , the separable
√
bound is tighter, ∣⟨C⟩∣ ≤ 2.
From a subset of the measured P⃗ in Experiment I and II, we obtain expectation values
 8.7. characterizing the quantum states 215

fidelity bounds
data
theory

-1
-180 -90 0 90 180
Rotation angle θ (deg)
Figure 8.16: Entanglement witness for entangled states. Experimental lower bounds
Bi (orange) on concurrence given by the optimal witnesses for Bell states WΨ+ (circles),
WΨ− (squares), WΦ+ (triangles), and WΦ− (crosses), and fidelity F to the ideal state (black cir-
cles) for experiment II. A maximum lower bound is reached by BΦ− (BΨ+ ) at θ = −90(+90)○ .
Solid curves are results of master equation simulations.

of four CHSH operators with L, L′ ∈ {X L , Z L } and R, R ′ ∈ {X R , Z R }. For experiment I

(figure 8.17), we find to within statistical error that ⟨CZXXZ ⟩ = ⟨XX⟩ − ⟨XZ⟩ + ⟨Z X⟩ + ⟨ZZ⟩
(squares) and ⟨CZXZX ⟩ = ⟨XX⟩+⟨XZ⟩−⟨Z X⟩+⟨ZZ⟩ (triangles) remain within one standard
deviation of the separable bound for all θ. Conversely, for the entangled states prepared
in experiment II (figure 8.18), ⟨C⟩ clearly oscillates well past the separable bounds. At
θ = ±45○ , a maximum value ∣⟨C⟩∣ = 2.61 ± 0.04 is reached. The agreement with theory and
√
proximity of this maximum ∣⟨C⟩∣ to the 2 2 upper bound [39] further demonstrate the
highly entangled states produced, and the ability of the joint measurement to determine the
√
degree of entanglement precisely. Here, ∣⟨C⟩∣ exceeds the separable state bound of 2 by
∼ 30 standard deviations. Furthermore, it also violates a Bell inequality by exceeding the
classical bound of 2 by ∼ 15 standard deviations. Locality and detection loopholes present
216 entanglement and joint readout

data
theory

2.82
2
1.41

-1.41
-2
-2.82
-180 -90 0 90 180
Rotation angle θ (deg)
Figure 8.17: CHSH for separable states. Experimental average value of CHSH operators
CZZXX (circles), CZXXZ (squares), CZXZX (triangles),√C XXZZ (crosses). For experiment I all
⟨C⟩ values stay within the separable state bounds ± 2 up to measurement noise (∼ 0.04).
Solid curves are results of master equation simulations.

in our system preclude a fundamental test disproving LHV. For Josephson phase qubits,
the detection loophole has recently been closed using the more traditional technique of
single-shot independent readouts [51]. We emphasize that we calibrate the measurement and
the gates but we do not specifically optimize for a maximum ⟨C⟩ value.

8.8 Chapter summary

In this chapter, we have extended two-qubit circuit QED using on-chip flux control. Using
the flux-bias lines, we are able to turn on a two-qubit σz ⊗ σz interaction which functions via
a coupling in the two-excitation manifold of the transmons. This permits the construction
of entangling c-Phase gates, which can be used to generate highly-entangled states such
as the Bell states. Furthermore, we have demonstrated a joint readout of superconducting
 8.8. chapter summary 217

data
theory

2.82
2
1.41

-1.41
-2
-2.82
-180 -90 0 90 180
Rotation angle θ (deg)
Figure 8.18: CHSH violation with entangled states. Experimental average value of CHSH op-
erators CZZXX (circles), CZXXZ (squares), CZXZX (triangles), C XXZZ (crosses). For experiment
II, max ∣⟨C⟩∣ = 2.61 ± 0.04. Solid curves are results of master equation simulations.

qubits using the microwave cavity as a single measurement channel to gives direct access to
qubit correlations. This readout is advantageous because it introduces the minimal number
of channels for qubit decoherence and is easy to model and calibrate accurately. The joint
readout represents a different strategy from that of individual qubit readouts, but is shown
to be a viable approach for precisely characterizing entangled states. In its present form,
this joint readout has the resolution to detect future improvements in two-qubit gates and
will be extendable to systems of three or four qubits. Applying this readout to analyze
highly-entangled states, we report the largest violation of CHSH inequalities in a solid-state
system. These results represent an advance in the ability to quantify the entanglement between
superconducting qubits. Furthermore, the possibility to measure multi-qubit parity operators
could be useful for quantum error correction, generating entanglement by post-selection
[148, 149] or fundamental tests of quantum contextuality [150, 151].
Next, in chapter 9 we put together all of the components we have discussed thus far, the
218 entanglement and joint readout

high-fidelity single-qubit gates from chapter 6 with the two-qubit c-Phase gate and joint
readout described in this chapter, to operate simple quantum algorithms. Specifically, we will
deal with the two-qubit Deutsch-Jozsa and Grover’s search algorithms, demonstrating the
ability of our circuit QED system to behave as a quantum processor.
 CHAPTER 9

Two-Qubit Algorithms

B
y harnessing superposition and entanglement of physical states, it has been proposed that
quantum computers could outperform their classical counterparts in solving problems
of technological impact, such as factoring large numbers and searching databases [12, 23]. A
quantum processor executes algorithms by applying a programmable sequence of gates to an
initialized register of qubits, which coherently evolves into a final state containing the result
of the computation. Simultaneously meeting the conflicting requirements of long coherence,
state preparation, universal gate operations, and qubit readout makes physical realizations of
quantum processors challenging.
Although few-qubit processors have already been demonstrated in nuclear magnetic
resonance [8–10], cold ion trap [152, 153] and optical [154] systems, a solid-state realization has
remained an outstanding challenge. Yet, over the last decade, superconducting circuits have
made considerable progress on all the requirements necessary for an electrically-controlled,
solid-state quantum computer. In the work for this chapter, we employ superconducting
qubits which have coherence times that have risen by three orders of magnitude to ∼ 1 µs, and
combine it with excellent state preparation and single-qubit gates as discussed in chapter 6 to
reach error rates [108, 116] of 1%. Furthermore, we have also demonstrated an engineered
two-qubit interaction tunable by two orders of magnitude on nanosecond time scales, capable
of generating entanglement to violate a CHSH inequality (chapter 8). Finally, we have also

219
220 two-qubit algorithms

shown that the microwave cavity can serve as a joint qubit readout, which exploits our
excellent single-qubt gates and circumvents poor readout fidelity (chapter 8).
In this chapter, we combine all of these achievements and demonstrate a two-qubit
superconducting processor capable of implementing the Deutsch–Jozsa (section 2.4.2) and the
Grover’s search (section 2.4.3) quantum algorithms. The quantum bus architecture combined
with the on-chip flux-bias lines provide the ability to couple, control, and measure the qubits.
By pulsing the qubit frequencies to an avoided crossing where a σz ⊗ σz interaction turns on
(section 4.3.3), we realize the two-qubit conditional phase (c-Phase) gate, theoretically treated
in section 2.3.2 and experimentally shown in section 8.3.1. Operation in the strong-dispersive
regime (section 3.4.3) [110] of cQED allows joint readout [127, 142] and detection of two-qubit
correlations through a single line (section 8.4). We will review the basics of the experimental
setup (section 9.1), although the algorithms are performed on the same sample cQED222
described in chapter 8. In section 9.2 we discuss the implementation of the Deutsch-Jozsa
algorithm for two superconducting qubits and in section 9.3 we demonstrate the simple
four-level Grover’s search.

9.1 Experimental details

The experimental sample and setup for implementing these algorithms are exactly the same
as those in the previous chapter. However, to understand the operation of our two-qubit
processor, it will be useful to recall the single excitations spectrum of figure 8.2. Again, the
two operation points will be locations I and II. Single-qubit gates and the joint readout will be
performed at location I while the two-qubit c-Phase gate will be implemented by flux-pulsing
into location II.
We have single-qubit rotations around the x and y axes of each qubit ({R x (θ), R y (θ)}, via
shaped microwave excitations, at our disposal, as well as the four different c-Phase gates (cU i j )
described previously. The pulse sequences for the algorithms are defined in Mathematica and
programmed into six channels across two arbitrary waveform generators (Tektronix AWG
520 and AWG 5014). We next review the two simple quantum algorithms (Deutsch-Jozsa
and Grover’s) and describe how we implement them in our circuit QED processor.
 9.2. deutsch-jozsa algorithm 221

9.2 Deutsch-Jozsa algorithm

The Deutsch–Jozsa (DJ) algorithm (section 2.4.2) is one of the simplest quantum algorithms to
implement and one of the quintessential examples of using quantum mechanics to determine
a quantity using fewer operations than a classical algorithm. It also represents a class of
deterministic quantum algorithms which provide an exponential speedup over classical
algorithms. The advantage of the DJ algorithm is explicitly described earlier in this thesis in
section 2.4.2. Here, we review a few of the basic concepts to understand how to implement it
in our two-qubit superconducting quantum processor.
To motivate the use of the DJ algorithm, recall the problem of finding out whether a coin
is fair, with heads on one side and tails on the other, or biased, with heads on both or tails on
both. The classical method for determining the answer requires that each side of the coin be
examined. However, this classical solution in fact gives too much information, as we not only
find out whether the coin is fair or not, but also exactly which sides are heads or tails. We can
instead employ the DJ algorithm on a register of two qubits and determine only information
about the nature of the coin in one examination step.
We can represent the four possible coins with four functions f that map one input
bit, x = 0, 1, representing the two sides of the coin, onto a single output bit, f (x) = 0, 1,
representing either heads or tails. There are only four such functions which take a single bit
to another single bit. Two of these functions are constant, or independent of the input bit
x, f0 (x) = 0, f1 (x) = 1, representing the fake coins. The other two functions are balanced,
f2 (x) = x, f3 (x) = 1−x, representing the fair coins. The DJ algorithm functions by performing
a bitwise addition of the functions evaluated on one of the qubits with the other qubit.
Specifically, as previously described in section 2.4.2, the operation of the function on one of
the qubits can result in a quantum phase kickback onto the state of the other qubit which
then permits determination of the quantity f (0) ⊕ f (1).

9.2.1 Breaking down the algorithm

The DJ algorithm programmed in our system is schematically represented in figure 9.1. We
identify the left qubit as the control qubit and the right qubit as the target qubit. Both qubits
in the register start off in the ground states, such that ∣ψ⟩0 = ∣0, 0⟩. The algorithm consists
of three main parts. The first block is simply a simultaneous single-qubit rotation on both
222 two-qubit algorithms

Deutsch-Jozsa algorithm State tomography

0 R y (π/) l l R y (π/) R x,y (, π , π)
Joint
Ui Dispersive
0 R y (−π/) r r ⊕ f i (l) R y (π/) R x,y (, π , π) Readout

Figure 9.1: Quantum circuit for DJ algorithm. The DJ algorithm is comprised of two stages
of single-qubit operations surrounding a two-qubit unitary U i . The two-qubit gates U i encode
the four functions of one bit to one bit, f0 (x) = 0, f1 (x) = 1, f2 (x) = x, and f3 (x) = 1 − x, as the
transformations ∣l , r⟩ → ∣l , r ⊕ f i (l)⟩ (⊕ denotes addition modulo 2). The encoding unitaries
π π/2 π π/2 −π/2 π
are U0 = I ⊗ I, U1 = I ⊗ R x , U2 = (I ⊗ R y R x )cU00 (I ⊗ R y ), and U3 = (I ⊗ R y R x )cU11 (I ⊗
−π/2
R y ), respectively. The final state tomography step involves 15 combinations of single-qubit
rotations for determination of ρ.

qubits to get into the superposition state

1
∣ψ⟩1 = (∣0, 0⟩ − ∣0, 1⟩ + ∣1, 0⟩ − ∣1, 1⟩) . (9.1)
2
The next step involves applying the unitary transformation which will perform the bit-
wise addition, ∣lr⟩ → ∣l(r ⊕ f i (l))⟩, for the four different functions f i . With single-qubit
rotations and two-qubit c-phase gates (cU i j ) at our disposal, we can identify the four unitary
transformations U i which engender the properties of the four possible functions f i :

U0 =1 ⊗ 1 (9.2a)
U1 =1 ⊗ R x (π) (9.2b)
U2 = [1 ⊗ R y (π/2)] cU00 [1 ⊗ R y (π/2)] (9.2c)
U3 = [1 ⊗ R y (−π/2)] cU11 [1 ⊗ R y (−π/2)] . (9.2d)

It is easy to understand U0 and U1 , as they are simply the identity, and a bit-flip of the
target qubit, respectively. These both do nothing to the state of the control qubit, and hence
at the end of the final step, where another π/2 rotation is applied, the state is measured as
∣1, 0⟩. Note that the target qubit is unaffected by U0 and U1 because the first −π/2 rotation
has placed it into an eigenstate of each unitary, with eigenvalue equal to 1.
In the case of the balanced unitaries, U2 and U3 , correspond to applying cNOT and
z − cNOT. It is trivial to build these gates from our c-Phase gates, requiring only single-qubit
 9.2. deutsch-jozsa algorithm 223

rotations on the target qubit in each case. Although neither U2 nor U3 directly alters the state
of the control qubit, a phase of −1 is attained as a result of its operation on the target qubit
superposition state. This phase in effect flips the superposition of the control qubit, such that
the final π/2 rotation in the last step takes the final state to ∣0, 0⟩.

9.2.2 Deutsch–Jozsa results

In our experiment, we run the algorithm for all four possible functions f and then perform
quantum state tomography on the two-qubit register. Although the operation of the algorithm
requires only the measurement of the control qubit (with the state ∣1⟩ corresponding to
constant and ∣0⟩ for balanced), state tomography permits us to observe the state of the target
qubit as well as any errors which may arise.
The two-qubit state tomography is performed as described in section 8.6. We also follow
the same prescription given earlier (section 8.6.1) for obtaining a density matrix through
maximum-likelihood estimation, ρML . Figure 9.2 shows the real part of the inferred ρML for
the four cases. All elements of the imaginary part of ρML are less than ∼ 0.05. For constant
functions f0 and f1 , shown in figure 9.2a–b, we find the final state ∣1, 0⟩ with a fidelity (as
given by Eq. (8.11)) of 0.94 and 0.95, respectively. For the balanced functions f0 and f1
(figure 9.2c–d), we measure the state ∣0, 0⟩ with fidelities of 0.92 and 0.85.
The fidelities to the constant functions are slightly higher than those of the balanced
functions due to to shorter length of pulse sequences. In the case of the constant functions,
there are only single-qubit operations, with each gate taking 8 ns and separated by 5 ns
buffering time, for a total of 34 ns. On the other hand, for the balanced functions, the unitary
step which implements f2 and f3 itself consists of 2 single-qubit rotations and a two-qubit
gate which takes 30 ns. Therefore, there are a total of 4 single-qubit rotations, the two-qubit
gate, and 5 ns buffer between all pulses resulting in a total algorithm time of 82 ns. The lower
fidelity for the balanced cases can be seen when repeating the algorithm multiple times. We
obtain statistics from 8 measurements of the four cases, finding fidelities to the ideal final
states of 0.93 ± 0.01, 0.93 ± 0.01, 0.909 ± 0.006, 0.841 ± 0.009.
Although the reduced fidelity for f2 and f3 are roughly consistent with the time it takes to
perform the algorithm and the coherence times of the system, a full characterization of the
unitary gates which generate the functions should be performed to understand the nature of
the errors. This work is left for the future.
224 two-qubit algorithms

(a) f (x) =  (b) f (x) = 

1 1

0.5 0.5
0 0
-0.5 11 -0.5 11
-1 10 -1 10
00 01 01 00 01 01
10 00 10 00
11 11
(c) f (x) = x (d) f (x) =  − x

1 1

0.5 0.5
0 0
-0.5 11 -0.5 11
-1 10 -1 10
00 01 01 00 01 01
10 00 10 00
11 11
Figure 9.2: Results for four cases in DJ algorithm. Real part of the inferred density matrix
ρml of the algorithm output in the four cases (imaginary elements of ρml are less than 0.05,
0.03, 0.05, 0.06, respectively). For the constant (balanced) functions f0 and f1 ( f2 and f3 ), ρml
shows a high fidelity to ∣1, 0⟩ (∣0, 0⟩), as expected. For the density matrices shown, the fidelities
to the ideal output states are F = 0.94, 0.95, 0.92, and 0.85.
 9.3. grover search algorithm 225

9.3 Grover search algorithm

The other quantum algorithm which we implement in our two-qubit processor is the Grover
search algorithm. It reflects a separate class of quantum algorithms which is probablistic in
nature–the correct solution can be found with high probability–and not deterministic like
the Deutsch-Jozsa algorithm which always gives the correct answer. We describe Grover’s
search algorithm in detail earlier in this thesis in section 2.4.3. Here we simply recall the basic
concepts behind the algorithm and discuss it in regards to our two-qubit implementation.
A classical search for a particular entry in an unordered list of N elements requires linear
time, or O(N) queries to the list. We can think of the problem as searching for a phone
number in a telephone book, and the classical search involves randomly choosing a name
from the list for the search. Grover’s algorithm, however, provides a method for search faster
√
than the classical case, and in the best case allows a quadratic speedup, or O( N) time.
With two qubits, it is possible to implement a Grover’s search with N = 4. Let the four
entries be represented by the set x = {0, 1, 2, 3} and there is a function f (x) which returns
the value 0 for all x except for x0 , where f (x0 ) = 1. Classically searching for x0 would require
on average 2.25 evaluations of the function f . However, using Grover’s algorithm, we can
simply use a single evaluation.

9.3.1 The oracle

In our implementation, we can identify the two-bit set as x ∈ {00, 01, 10, 11}, being the labels
of our four computational basis states. Grover’s algorithm allows us to determine the x0 at
which f (x0 ) = 1 through a single call to an oracle O which has encoded f (x) into a quantum
phase,

O ∣x⟩ = (−1) f (x) ∣x⟩ . (9.3)

Hence, for all x ≠ x0 , we simply return the same state, whereas for the target x0 , we pick up
a −1 phase difference. An oracle which realizes the relation of Eq. (9.3) is the c-Phase gate
(section 2.3.2 and section 8.3.1). Specifically, for finding any of the 4 x = i j, the oracle to be
used is then O = cU i j .
eplac
226 two-qubit algorithms

Grover algorithm State tomography

0 R y (π/) R y (π/) R y (π/) R x,y (, π , π)
Joint
a b O c cU e f Dispersive
R y (π/) R y (π/) R y (π/) R x,y (, π , π) Readout
0
d
Figure 9.3: N=4 Two-qubit Grover algorithm schematic. Concatenation of single-qubit and
c-Phase gates implementing one iteration of Grover searching. Without loss of generality, we
π π/2
have replaced the Walsh–Hadamard transformations W = R x R y in the usual description
π/2
of the algorithm [12, 23] with R y rotations in order to eliminate 6 single-qubit rotations
and complete the sequence in 104 ns. The orange box is the oracle O = cU i j that encodes the
solution x0 = i j to the search problem in a quantum phase. Note that the first half of the
algorithm is identical to the entangling sequence in figure 8.7 for generating Bell states, while
the second half is essentially its mirror image.

9.3.2 Breaking down the algorithm

The basic Grover’s algorithm consists of three main parts (section 2.4.3): superposition state
initialization, oracle call to mark the targeted state with a phase, conditional phase to amplify
the amplitude of the targeted state. Whereas an N-entry Grover’s search can require multiple
iterations of the last two parts, in our 4 state case, we will only need to run each step once.
Our sequence for implementing Grover’s algorithm is schematically illustrated in figure 9.3.
Both qubits start off in the ground state, ∣ψ⟩0 = ∣0, 0⟩. The algorithm begins with the
initialization of a superposition of all 4 possible elements in the set is performed using
simultaneous single-qubit π/2 rotations on both the left and right qubit, R = R Ly (π/2) ⊗
R Ry (π/2). This results in the maximal superposition state

1
∣ψ⟩1 = (∣0, 0⟩ + ∣0, 1⟩ + ∣1, 0⟩ + ∣1, 1⟩) . (9.4)
2
Next, the oracle operator is applied, which in our case is any one of the four c-Phase gates,
cU i j , depending on which of the four entries we would like to search for. For illustrative
 9.3. grover search algorithm 227

purposes, let us search for the third entry i j = 10, so that we apply

⎛1 0 0 0⎞
⎜ ⎟
⎜0 1 0 0⎟
O = cU10 = ⎜
⎜
⎟.
⎟ (9.5)
⎜0 0 −1 0⎟
⎜ ⎟
⎝0 0 0 1⎠

After this step, the state of the two-qubit register is then

1
∣ψ⟩2 = (∣0, 0⟩ + ∣0, 1⟩ − ∣1, 0⟩ + ∣1, 1⟩) , (9.6)
2
where the oracle has done its job to mark the solution with a phase in front of the relevant
searched entry, ∣1, 0⟩.
At this point of the algorithm, the solution has already been encoded into the quantum
register. However, to be able to determine the solution, a deterministic measurement on
the qubits will need to be performed. A measurement at this stage would simply give a
probabilistic result of one of the four computational basis states. The nature of the third
part of the algorithm is to increase the probability that the measurement will return the
targeted entry. This is performed by applying the combination of another set of simultaneous
single-qubit π/2 rotations, R, with a conditional phase gate cU00 , followed by R again. We
can see that after the first RcU00 , we are then in the state
1
∣ψ⟩3 = (∣0, 0⟩ − ∣0, 1⟩ + ∣1, 0⟩ − ∣1, 1⟩) . (9.7)
2
These two pulses combined apply a phase shift of −1 to all states which are orthogonal to the
state ∣0, 0⟩. The resulting state is simply a separable state, and the final application of R, in
fact leaves us in the targeted state ∣1, 0⟩. The combination of RcU00 R in effect inverts the state
around the mean and has transferred the phase in front of the ∣1, 0⟩ in ∣ψ⟩2 into an amplitude
which can be detected in a computational basis state measurement.

9.3.3 Grover results and debugger

Experimentally, we simply concatenate our single-qubit and two-qubit gates as prescribed in
the schematic figure 9.3. The single-qubit gates used here have σ = 2 ns with a total width of
4σ. A buffer time of 5 ns is left in between all stages of pulses to ensure complete turn-off.
The first applied conditional phase gate which serves as the oracle is tuned up as prescribed
228 two-qubit algorithms

a Grover algorithm State tomography

0 R y (π/) R y (π/) R y (π/) R x (π)

cU s cU Readout

0 R y (π/) R y (π/) R y (π/) R x (π/)

b
IL

QL

IR
s
QR

VL

VR

M
0 20 40 60 80 100 120
time (ns)
Figure 9.4: Microwave and flux pulses for Grover algorithm. (a) An example sequence, ex-
ecuting the Grover search algorithm with oracle O = cU10 and measuring M13 = −β1 σzL +
β2 σ yR − β12 σzL ⊗ σ yR . (b) Illustration of the microwave and flux pulses realizing the operations
directly above. All microwave pulses implementing the x- and y-rotations have Gaussian
envelopes, with standard deviation σ = 2 ns, truncated at ±2σ. The rotation axis is set using
I-Q (vector) modulation (chapter 5) and the rotation angle is controlled by pulse amplitude.
Flux pulses implementing c-Phase gates have three tuning parameters: the amplitude A of VR ,
the amplitude B of 1 ns ledges at the beginning and end of the VR -pulse, and the amplitude C
of VL (section 8.3.1). The flux pulse duration is fixed at 30 ns.
 9.3. grover search algorithm 229

in section 8.3.1. The voltage levels of the flux bias pulses on both the left and right flux bias
lines are found for each of the four different conditional phase gates.
The second applied conditional phase gate, however, does not behave the same way as the
first conditional phase gate. When concatenating flux pulses, there tends to be a ‘memory’
effect which causes a phase shift on the qubits due to the first pulse. The second pulse is
affected in the same way by any of the flux pulses used for the first c-Phase, so fortunately it
does not affect the performance of the algorithm. Rather, the voltage levels of the second
c-Phase needs to be tuned independently of the first. We employ experiments much like the
tune-up of the first c-Phase, but include a flux pulse with a duration and amplitude equal to
that of the first c-Phase.
Therefore, when we run the algorithm according to the prescription, the second c-Phase
gate uses a different set of flux bias voltages from the first. Nonetheless, the algorithm still
functions as it should, and we can investigate its action every step of the way. We employ
a debugging technique by interrupting the algorithm after each step and performing state
tomography. Figure 9.5 shows the step-by-step breakdown of our Grover’s search algorithm.
We perform two-qubit state tomography (section 8.6) using our joint readout (section 8.4) at
6 different stages of the algorithm, and map out the behavior of the two qubits along the way.
Maximum-likelihood estimation is used to obtain the density matrix ρml , which we plot in
the standard computational basis.
The first density matrix figure 9.5a shows the initialization of the two-qubits in the ground
state ∣0, 0⟩. Here, the fidelity is 0.99 to the ideal ground state. Next, the single-qubit π/2
rotations place us in the maximal superposition state (9.4), signified in the density matrix
with all the bars raised to a value ∼ 0.25, figure 9.5b. This preparation gives a fidelity to the
ideal state of 0.99. Then, the oracle is applied with flux pulses, here chosen to be O = cU10 ,
and leaving us in the entangled state (9.6). This density matrix figure 9.5c looks similar to
the previous one, except the searched entry i j = 10 is now marked with all the non-diagonal
bars along the 10 row and column being negative (fidelity is 0.83). We can see that this is
maximally entangled state as we can perform a single-qubit operation (R y (π/2)) on the left
qubit, and end up with figure 9.5d, which is the Bell state ∣Ψ+ ⟩. The fidelity to this Bell state is
0.88. Next, a single-qubit gate is applied to the right qubit, followed by the second c-Phase
(cU00 ), resulting in the state given in (9.7). This second c-Phase has taken the phase from the
oracle and re-distributed it into a unique separable state (fidelity is 0.91), figure 9.5e, which
can be transformed with the simultaneous single-qubit rotation R into the final solution state
∣1, 0⟩, shown in figure 9.5f. The fidelity to the final state is found to be 0.82.
230 two-qubit algorithms

(a) (b) (c)

1
0.5
0
-0.5 11
-1 10
00 01
01 00
10
11
(f ) (e) (d)

Figure 9.5: Implementing Grover’s algorithm. Real part of ρml obtained by state tomography
after each step of the algorithm with oracle O = cU10 . Starting from ∣0, 0⟩ (a), the qubits are
simultaneously rotated into a maximal superposition state (b). The oracle then marks the
π/2
solution, ∣1, 0⟩, by inverting its phase (c). The R y rotation on QL turns the state into the Bell
π/2
state ∣Ψ+ ⟩, demonstrating that the state is highly entangled at this stage (d). The R y rotation
on QR produces a state identical to (c) (data not shown). The application of cU00 undoes the
entanglement, producing a maximal superposition state (e). The final rotations yield an output
state (f) with fidelity F = 82% to the correct answer, ∣1, 0⟩.

Similar performance is obtained for the other three oracles and shown in figure 9.6. The
reduction in fidelity of ∼ 15 − 20% is consistent with the coherence times of the two qubits and
the 104 ns it takes to perform the entire algorithm and measurement sequence (3 single-qubit
rotations, 2 c-Phase gates, another single-qubit rotation for state tomography).
However, the issue with needing to re-tune the second c-Phase gate makes the current
implementation difficult to scale up to more and more complex algorithms. Investigations
are presently being carried out to ascertain the nature of the flux pulse memory and creative
pulse shaping schemes for combatting it are being developed. Also, the fidelity to the states at
different points of the Grover debugger is not a simple monotonic function of the total gate
time. Rather, there is some variation with respect to the fidelities after specific stages. We do
 9.4. chapter summary 231

O = cU O = cU
(a) (b)
1 1

0.5 0.5
0 0
-0.5 11 -0.5 11
-1 10 -1 10
00 01 01 00 01 01
10 00 10 00
11 11

O = cU O = cU

(c) (d)
1 1

0.5 0.5
0 0
-0.5 11 -0.5 11
-1 10 -1 10
00 01 01 00 01 01
10 00 10 00
11 11
Figure 9.6: Grover fidelity for all choice of oracles. The Grover’s algorithm is run for all
four choices of oracles O = cU00 , cU01 , cU10 , cU11 , finding fidelities of 0.81, 0.80, 0.82, 0.81,
respectively.

not yet understand the reason for this, but further characterization of the two-qubit gate and
combinations of two-qubit and single-qubit gates via process tomography or randomized
benchmarking techniques could clarify the picture greatly. This work is left for the future.

9.4 Chapter summary

In summary, we have demonstrated two-qubit quantum algorithms using a superconducting

circuit. The incorporation of local flux control and joint-dispersive readout into cQED, to-
gether with a tenfold increase in qubit coherence over previous two-qubit devices, has enabled
on-demand generation and detection of entanglement and the implementation of the Grover
and Deutsch–Jozsa algorithms. As the first demonstration of such quantum algorithms in a
solid-state system, we can look towards more interesting and complex quantum information
232 two-qubit algorithms

processing using circuit QED. Specifically, the present architecture can be immediately ex-
panded to several qubits with controllable σz ⊗ σz interactions between nearest-frequency
neighbors, placing within reach the generation of Greenberger–Horne–Zeilinger states and
the exploration of basic concepts of quantum error correction [12, 23]. The final chapter of
this thesis will suggest some more future directions for our quantum information processor
and provide a little broader perspective on quantum computing as a burgeoning field of
research.
 CHAPTER 10

Conclusions and Future Work

T
he rudimentary superconducting quantum processor presented in this thesis represents
a major first step towards the achievement of a larger scale quantum computer. Over
the past ten years superconducting circuit-based quantum computing has gone from a
few Rabi oscillations that lasted just nanoseconds to the programming of simple quantum
algorithms. Yet, moving forward, there is still a formidable challenge of how to build on what
has been demonstrated to push the envelope both on the technological aspects of a quantum
computer as well as the physics of quantum information processing. The rapid progress of
superconducting qubit based quantum computing has been due in large part to the already
well-developed protocols for NMR, photon, and trapped-ion quantum computing systems.
The immediate task at hand of scaling the superconducting system to more than two qubits
will most likely be no different, as it takes merely minutes to scour through the literature to
find three and four qubit experiments already implemented to build a list that will take more
than two years of work to accomplish. Regardless, these are necessary steps for demonstrating
that the superconducting-circuit based quantum computing system is viable and possibly a
more attractive option further down the road.
Here, some reflections on extending the work presented in this thesis are given, broken
down into ideas for improving the control over single and two-qubit operations (section 10.1)
and a few proposed experiments for expanding past two qubits (section 10.2). Finally, an

233
234 conclusions and future work

outlook for the longer term prospects of scaling quantum information systems is given
(section 10.3).

10.1 Improving one and two-qubit operations

As mentioned previously in this thesis (chapter 6), the ultimate goal of a fault-tolerant
quantum computer will require considerably lower error rates (10−4 − 10−5 ) than the 10−2
achievable at this time. Furthermore, the current two-qubit c-Phase gate lags even further
behind with the generated entangled states achieving state fidelities ∼ 90%. There is certainly a
lot of room for improvement and working towards the fault-tolerant threshold is an important
direction. Here, we highlight a few possible directions.

10.1.1 Longer coherence times

Perhaps the most critical component towards improving both single and two qubit operations
is obtaining longer coherence times. It is safe to say that all the gate fidelities would improve
significantly with another order of magnitude of improvement from our current 1 − 2 µs
range. As briefly discussed in section 3.5.1, the transmon qubits currently have relaxation
times which are well-understood through the multi-mode Purcell effect up to the limit of
an intrinsic Q ∼ 50, 000 − 70, 000. At this stage, it is unclear as to what is the direct cause of
this intrinsic limit, which is seen not only with the transmon qubits used in our circuit QED
architecture, but in other superconducting qubit experiments as well [155, 156].
There have been a number of proposed loss mechanisms, including decaying into two-
level systems on the surfaces of the superconductor [157], dissipation into the dielectric
substrate [155], trapped surface vortices, and decay into non-equilibrium quasiparticles
near the Josephson junctions. Which, if any, of these are the reason for the intrinsic loss is
currently under investigation. Specifically, there are current efforts in our group to examine
the quality factors of substrate materials (such as sapphire, or silicon-dioxide). This can
be investigated by making a very good high-Q factor single-photon cavity, and finding out
whether it can be spoiled by including dielectric slabs. Another avenue is to specially engineer
the superconducting films around the Josephson-junctions of the superconducting qubits in
order to reduce stray non-equilibrium quasiparticles from interacting with the junctions.
Outside of the intrinsic loss, in the circuit QED system, the qubit relaxation times are
governed by the multi-mode Purcell effect. In terms of T1 of the qubit, it would seem like
 10.1. improving one and two-qubit operations 235

the best result is to use narrower cavities, and decreasing the spontaneous emission decay.
However, this is not optimal for readout, as it can result in a very poor signal-to-noise ratio
due to fewer signal photons being collected during a qubit T1 [53]. A solution to avoid the
Purcell-limited qubit decay is to employ a carefully microwave-engineered addition to the
cavity which changes the real part of the impedance at a specific frequency [82]. This idea of a
“Purcell filter” would then allow a qubit to have longer relaxation times while also maintaining
a fast readout.

10.1.2 Better qubit operations

Although improved coherence times would undoubtedly make single-qubit gates better, we
can also build on the work in section 6.4, where we employ derivative-pulse shaping to
improve the gate fidelity. As discussed in section 4.2.3, the anharmonic spectrum of the
transmon qubit can result in leakage out of the computational basis at shorter gate lengths.
Pulse shaping techniques can be used to mitigate this effect. Although the experimental work
in section 6.4 does not prove that the error mechanism is due to leakage to the second excited
state, the pulse-shaping does drastically improve the gate fidelity for shorter gate lengths.
A true test of whether the error mechanism is the anharmonicity would be to test for the
population of the ∣2⟩ state.
Nonetheless, as we start to add more and more qubits, the Hilbert space becomes quite
large and even weak residual cavity-mediated interactions can be a source of error. It is
thus important to actually determine whether a single-qubit is behaving as an uncoupled
single-qubit should. A novel way to test this would be to use the randomized benchmarking
technique to extract the average gate fidelity of a single-qubit, but while a residual coupling
to another qubit is still on. We can see from section 8.3 that even when the qubits in that
experiment were detuned by over ∼ 2 GHz, a residual σz ⊗ σz interaction still persisted on
the order of ∼ 1 − 2 MHz. An experiment which attempts randomized benchmarking on one
qubit, while performing arbitrary operations on the other qubit, would be a good metric
for how the average fidelity degrades due to the presence of this always-on interaction. To
advance this even further, one could imagine repeating the same experiment but for different
pulse-shapes to see if the effect of the coupling could be minimized through optimal control
techniques.
Other improvements in single-qubit operations could be sought through using techniques
such as optimized dynamical decoupling [158] to suppress specific forms of error. These
236 conclusions and future work

techniques have already begun to be used in trapped-ion [159] and electron spin systems
[160]. However, the extension of such protocols to improve the performance of arbitrary gate
sequences is still a topic of ongoing theoretical research.
Whereas in this thesis we have detailed the importance of single-qubit gate character-
ization (chapter 6), we have yet to accomplish the same level of rigor with the two-qubit
entangling gate demonstrated in chapter 8. Specifically, we may ask the same questions as in
the single-qubit gate case: what are the gate fidelity and process fidelity associated with the
c-Phase?
To characterize two-qubit gates, we may again consider two techniques used with single-
qubit gates, namely process tomography and randomized benchmarking. As discussed in
chapter 6, process tomography can theoretically give a lot of information about the underlying
gate, and would result, in the two-qubit case, with a 16 × 16 process matrix χ. The protocol
would be very similar to that given in figure 6.9, with the preparation of at least 16 starting
states to span the two-qubit Hilbert space, followed by the action of the two-qubit gate,
and then ending with state tomography for all 16 cases. However, the same caveats for
QPT being a good gate metric in the single-qubit case arise for the two-qubit case as well.
Specifically, generating the starting states and the state tomography steps both involve single-
qubit rotations, which themselves will introduce errors. However, QPT could still be useful for
determining certain systematic errors in the single or two-qubit gates applied on the system,
and it reflects an important step for comparison purposes with other quantum computing
systems, such as NMR, trapped-ions, and photons, in which the protocol has already been
achieved and two-qubit gate benchmarks show fidelities of ∼ 90 − 95% [35, 114, 115].
Randomized benchmarking can be extended to the two-qubit system as well, to obtain
an average estimate of the gate fidelity. Details for how to implement RB including two-qubit
gates are given in Ref. [119], where RB is used on an NMR system. One main difference from
the single qubit case is the inclusion of the cNOT gate, which is itself in the set of two-qubit
Clifford group generators and can give a fully depolarized noise channel [161]. Similar to the
single-qubit case, the extracted average gate errors could be compared with theory taking
into account the relevant time-scales of the system. It would also open up the possibility of
testing optimal control pulse shaping on the two-qubit gates, just as we have begun to do
with the derivative pulse-shaping technique on single-qubit gates (section 6.4).
One more intriguing idea within two-qubit experiments is to characterize the two-qubit
entangled state space. Numerous experiments have now been carried out detailing the
single-qubit Hilbert space quite well, from measuring the relaxation and coherence times to
 10.2. more qubits in circuit qed 237

determining gate and state fidelities. However, it would be interesting to obtain similar metrics,
but for the Hilbert space spanned by two entangled states. For example, in section 8.6.3, we
were able to rotate a single-qubit after preparing an entangled state. This action did not change
the entanglement of the state, but did perform rotations within a pseudo-Bloch sphere where
the poles are two of the Bell states. It is fair to then ask what the decoherence and relaxation
properties are in this Bell basis. For example, we can imagine a ‘hyper-Ramsey’ experiment,
by generating a superposition state of two Bell states (i.e. by applying a π/2 pulse in the Bell
basis), and allowing the state to undergo free precession, before undoing the superposition
to determine the decoherence time. A further motivation for understanding this sub-qubit
within two entangled qubits is the dark state, as discussed in section 7.2.2. Recall that the Bell
√
state ∣Φ+ ⟩ = (∣0, 1⟩ + ∣1, 0⟩)/ 2 could not be directly driven through the microwave cavity,
and hence does not couple to the environment through the cavity. Therefore, that state could
possibly be part of a ‘decoherence free sub-space’ (DFS) [162], with many further applications
for errorless quantum computing. Nonetheless, understanding the dynamics of the dark state
or the super-radiant state would be critical for determining if indeed there is an accessible
DFS.

10.2 More qubits in circuit QED

The list for experiments with only two-qubits could probably go on, but alas to build a
quantum computer scaling past two is a necessity. Fortunately, the current circuit QED
design is not limited to just 2 qubits coupled to a single microwave CPW resonator. Even
by continuing to drive the λ/2 resonance, it is possible to place up to 4 qubits, as shown in
figure 10.1. Each qubit is independently flux tunable via its own flux line. Such samples have
already been made and are now in the process of being tested and characterized by Leonardo
DiCarlo and Matt Reed.
With the independent flux tuning, it will be simpler to use only three out of the four qubits
first, and detune one of the qubits far away from any interactions with the cavity or with the
other qubits. The simplest experiments to aim for are to repeat the characterization of the
joint readout, but now for a three-qubit system. Operating in the strong dispersive regime of
3 qubits, there would now be 8 different cavity shifted frequencies. Similar protocols to those
presented in this thesis involving pre-rotations could be used to construct the three-qubit
Pauli set P⃗ or the density matrix ρ.
 238 conclusions and future work

Figure 10.1: Four qubit circuit QED sample. New four qubit circuit QED samples places two
qubits on each end of the resonator. The device is now a 6 port device, as there are now 4
flux-bias lines for each of the qubits, in addition to the input and output ports of the resonator.

Generating two-qubit entangled states will be done in exactly the same way. Recall that
the controlled z interaction used to generate the c-Phase gate (section 4.3.3) functions by
adiabatically pulsing into an interaction between the ∣1, 1⟩ and ∣0, 2⟩ states. For this to occur,
the two transmon ground to first excited state have to be nearest neighbors in frequency
space∗ . Therefore, if we imagine starting with three qubits such that ω(1) > ω(2) > ω(3) , we
would like to have access to a c-Phase gate between qubits 1 and 2, and another gate between
2 and 3. In that situation, the protocol for generating a maximally entangled three-qubit
√
state, such as the GHZ state given by ∣GHZ⟩ = (∣000⟩ + ∣111⟩)/ 2, is quite simple and shown
schematically in figure 10.2a.
Now having access to three qubit entanglement opens up a wealth of possible experiments.
One interesting path is to implement simple quantum error correcting codes. Quantum
error correction is an an important step for quantum computing, as it can be used to protect
the information stored in qubits from errors such as decoherence or systematic gate errors
[12]. Most quantum error correcting codes involve spreading out the information of a single-
qubit across the entangled state of a register of qubits. One of the simplest codes employs
3 qubits, and can be used to correct for dephasing errors [163]. It was first experimentally
demonstrated in NMR [164] and trapped ions [165]. Recently, the protocol has been proposed
for implementation in superconducting qubits [166], employing exactly the same conditional
phase gates which we have access to in our current setup. A schematic of the required gate
sequence is shown in figure 10.2b. One important thing to note is the final step requires either

∗
Optimized pulse-shaping of the flux could possibly allow a Landau-Zener type transition across qubit transition
frequencies.
 10.2. more qubits in circuit qed 239

(a) 0 R y (π/)


∣GHZ⟩ = √ (∣⟩ + ∣⟩)
R y (π/) 
0

0 R y (π/)

(b) ψ R y (π/) R y (π/) ψ

0 R y (π/) E R y (π/)

0 R y (π/) R y (π/)

Figure 10.2: Protocols for GHZ and 3-qubit code.(a) The GHZ state is generated by first
applying π/2 rotations onto each of the three qubits, generating a maximal superposition
state. Then, that is followed by two successive c-Phase gates, first between the first two
qubits, and then between the second and third qubits. (b) The three-bit code allows for the
correction of the state on one qubit. First a set of cNOT gates are used to encode the state
of the qubit into the three qubit register. That is followed by single-qubit π/2 rotations and
then the error process E. The state of the qubit is de-encoded and a final Toffoli gate is used
to leave the corrected qubit in its initial state.

performing a Toffoli gate [25] or a final single-qubit rotation conditioned on the measurement
of two of the qubits.
The experimental demonstration of the Toffoli gate would be a key hallmark to achieve
in itself, as it is an archetypal three-qubit gate, valuable in many more complex quantum
algorithms. It can be especially useful if it achieves a task with using fewer resources than the
equivalent one and two-qubit gate decomposition. Although certainly it could be built from
one and two-qubit gates, a novel direction with transmon qubits in circuit QED is to dive
into the spaghetti of energy levels and search for a three-qubit interaction along the lines of
the σz ⊗ σz interaction which we used to generate our c-Phase gate.
Here again, the list of experiments can go on and on, but hopefully this has provided
some perspective for interesting three qubit experiments. Scaling past 3 and 4, there are even
240 conclusions and future work

more error correcting codes to attempt such as the Steane code with 7 qubits [163], as well as
quantum teleportation experiments [167–170] and even quantum simulation of Hamiltonians
[171, 172].

10.3 Quantum information outlook

For a view towards the future of superconducting qubits, it is important to recognize the
rapid progress over the past ten years, getting from coherence times of just a few nanoseconds,
to being able to perform hundreds of operations and even implementing basic quantum
algorithms. To strike an analogy with the classical computer which we started this thesis
discussing, the current development of our two-qubit superconducting quantum processor
at Yale could be akin to the four-bit Intel 4004 microprocessor in 1971. Subsequently, it took
about ten years to transition to IBM’s PC in the 1980s, implementing the 16-bit Intel 8088
microprocessor. By 1982-83, the IBM personal computer became an industry standard and
almost all leading manufacturers switched into making products that would be compatible
with IBM’s computer. At Yale, we are now progressing to 3 and 4 superconducting quantum
processors and it would be interesting to see how far we can get in the next ten years.
Suppose the experiments outlined here go according to plan: superconducting coherence
times increase by another two orders of magnitude; gate operations reach the fault-tolerant
threshold; more complex quantum algorithms and error correcting codes are implemented.
Then what? From a quantum engineering point of view, that would represent a shift away
from understanding just simple quantum information processing very well, towards scaling
the system up and making a quantum computer that might be useful for practical purposes.
In reference to the classical computers, the problem becomes one of engineering the quan-
tum integrated circuit for scalability. At that point, progress could be as precipitous as the
subsequent development of computing from the 8088 microprocessor.
How might we envision this quantum computer in terms of superconducting qubits in the
circuit QED architecture? First, we would have to think about how to interconnect ever more
qubits. With error correcting codes requiring ∼ 5−7 qubits to give a single fault-tolerant qubit,
we could envision that one microwave resonator with these 5 − 7 qubits coupled to it would
be used as a single logical qubit. From there, a new challenge emerges in terms of how to
couple this cluster of qubits to another similar cluster of qubits. Perhaps a more complicated
architecture involving coupling between multiple resonators is necessary. Furthermore, an
actual implementation would probably require operations to be run massively in parallel on
 10.3. quantum information outlook 241

many qubits at once, and then to have interactions and information transfers switchable to
communicate between different clusters. The hardware to control such a behemoth operation
would need to be very well-developed and planned as well.
For now, we should be excited about our two-qubit processor and immediate next set of
experiments. Though there are still numerous hurdles to get over before being able to scale
towards a large quantum computer, the ever-growing research emphasis on superconducting
qubits and the rapid progress over the past decade might not make the wildest quantum
computing dreams that far out of reach.

ñ
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 APPENDIX A

Mathematica code for microwave pulse generation

C
ode for generating pulse sequences for the various experiments of chapters 6 to 9.
The pulse sequences are generated as .pat files, which are uploaded into the arbitrary
wave form generators, either the AWG520 or the AWG5014. Each experiment corresponds
to running a sequence of many different .pat files. Generating the .pat files is done in
Mathematica; pulse shapes are defined using various functions (gaussian, square, hyperbolic
tangent, etc.). The DAC levels of the AWG are governed by calibration experiments. The first
section of the code defines the names of the various types of pulses, whether they be for qubit
drive or flux-bias, as well as the relevant timings and delays. The function PulseIdentify[]
serves as a look-up table for all of the different drive and flux pulses which we use. Then, we
give various pulse shape functions, as well as a function, GetNewSeq[], for concatenating
many different pulse types. The section ‘Calibration Sequences’ gives a number of experiments
for tuning-up qubit drive pulses and flux pulses. Then, we also provide the code for generating
randomized benchmarking sequences, single-qubit process tomography, and two-qubit state
tomography.

259
260 mathematica code: pulse generation

Pulse Parameters and Names

ü Define variables for pulses

H* DAC Offsets for defining zero*L
H* QUBIT 1*L

H*QUBIT 2 *L
DACoffsetLX = 512; DACoffsetLY = 512; DACoffsetLZ = 0;

H*Define the amplitudes for all pê2 pulses HX,Y,+,-L*L

DACoffsetRX = 512; DACoffsetRY = 512; DACoffsetRZ = 0;

H* QUBIT 1*L

H* QUBIT 2*L
ampLX90m = 342; ampLY90m = 342; ampLX90p = 682; ampLY90p = 682;

H*Define the amplitudes for all p pulses*L

ampRX90m = 342; ampRY90m = 342; ampRX90p = 682; ampRY90p = 682;

H*QUBIT 1*L

H*QUBIT 2*L
ampLXm = 172; ampLYm = 172; ampLXp = 852; ampLYp = 852;

ampRXm = 172; ampRYm = 172 ; ampRXp = 852 ; ampRYp = 852;

H*Flux pulse default widths*L

H*Flux pulse default amplitudes, Step 1 of Grover*L

fwidthL = 0; fwidthR = 0;

ampC0L = 0; ampC1L = 0; ampC2L = 0; ampC3L = 0;

H*Flux pulse default offsets, Step 1 of Grover*L

ampC0R = 0; ampC1R = 0; ampC2R = 0; ampC3R = 0;

offsetC0L = 0; offsetC1L = 0; offsetC2L = 0; offsetC3L = 0;

H*Flux pulse default amplitudes, Step 2 of Grover*L

offsetC0R = 0; offsetC1R = 0; offsetC2R = 0; offsetC3R = 0;

ampC0Lb = 0; ampC1Lb = 0; ampC2Lb = 0; ampC3Lb = 0;

H*Flux pulse default offsets, Step 2 of Grover*L

ampC0Rb = 0; ampC1Rb = 0; ampC2Rb = 0; ampC3Rb = 0;

offsetC0Lb = 0; offsetC1Lb = 0; offsetC2Lb = 0; offsetC3Lb = 0;

offsetC0Rb = 0; offsetC1Rb = 0; offsetC2Rb = 0; offsetC3Rb = 0;
H* Global pattern times *L H* in ns*L

H*Delays*L H* in ns*L
fixedPointCh1 = 5000; cycleLength = 9000; MeasPulseLength = 3000;

SpecPulseDelay = 0; MeasPulseDelay = -40;

FluxPulseDelay = 20; SpecBufferDelay = -65;

H* Pulse buffer times *L

AWG520Delay = 468; triggerWidth = 100; delay = 5; fluxdelay = 5;

bufferSS = 3; H*spec followed by spec*L

bufferS = 5; bufferF = 5;

bufferFS = 3; H*flux followed by spec*L

bufferSF = 3;H*spec followed by flux*L

H*Default Spec pulse parameters *L

bufferFF = 0;H*flux followed by flux*L

GaussWidth = 3; numSigmas = 4;

H* Actual pulse length is numSigmas*GaussWidth *L

SpecPulseLength = GaussWidth

SpecBufferMargin = 10;
 a. mathematica code: pulse generation 261

ü Pulse sequence parse function

The following function is used as a lookup table of different kinds of pulses, which can be p or p/2 pulses on
either qubit, as well as flux pulses for z-rotations. Depending on the pulse, it assigns certain amplitudes and
offsets.
PulseIdentify@test1_D := Module@8phaseL, ampL, phaseR, ampR, ampLZ,

H* Set default values *L

widthLZ, offsetLZ, ampRZ, widthRZ, offsetRZ, spec, flux, Ssigma<,

phaseL = 0; H* 0 for X, pê2 for Y*L

ampL = DACoffsetLX; phaseR = 0; ampR = DACoffsetRX;
ampLZ = DACoffsetLZ; widthLZ = 0; offsetLZ = 0;
ampRZ = DACoffsetRZ; widthRZ = 0; offsetRZ = 0;
spec = 1; flux = 0;

H*Default pulse case will return Identity, Id*L

Ssigma = SpecPulseLength;

H* Single Qubit L Pulses *L

If@test1 ã LX90p, 8phaseL = 0, ampL = ampLX90p<D;
If@test1 ã LX90m, 8phaseL = 0, ampL = ampLX90m<D;
If@test1 ã LY90p, 8phaseL = p ê 2, ampL = ampLY90p<D;
If@test1 ã LY90m, 8phaseL = p ê 2, ampL = ampLY90m<D;
If@test1 ã LXp, 8phaseL = 0, ampL = ampLXp<D;
If@test1 ã LXm, 8phaseL = 0, ampL = ampLXm<D;
If@test1 ã LYp, 8phaseL = p ê 2, ampL = ampLYp<D;
If@test1 ã LYm, 8phaseL = p ê 2, ampL = ampLYm<D;
H* Single Qubit R Pulses *L
If@test1 ã RX90p, 8phaseR = 0, ampR = ampRX90p<D;
If@test1 ã RX90m, 8phaseR = 0, ampR = ampRX90m<D;
If@test1 ã RY90p, 8phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã RY90m, 8phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã RXp, 8phaseR = 0, ampR = ampRXp<D;
If@test1 ã RXm, 8phaseR = 0, ampR = ampRXm<D;
If@test1 ã RYp, 8phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã RYm, 8phaseR = p ê 2, ampR = ampRYm<D;
H* Composite Single Qubit LR Pulses *L

8phaseL = 0, ampL = ampLX90p, phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LX90pRX90p,

If@test1 ã LX90mRX90p, 8phaseL = 0, ampL = ampLX90m,

If@test1 ã LY90pRX90p, 8phaseL = p ê 2, ampL = ampLY90p,

phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LY90mRX90p, 8phaseL = p ê 2, ampL = ampLY90m,

phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LXpRX90p, 8phaseL = 0, ampL = ampLXp, phaseR = 0, ampR = ampRX90p<D;

phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LXmRX90p, 8phaseL = 0, ampL = ampLXm, phaseR = 0, ampR = ampRX90p<D;

8phaseL = p ê 2, ampL = ampLYp, phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LYpRX90p,

If@test1 ã LYmRX90p, 8phaseL = p ê 2, ampL = ampLYm,

If@test1 ã LX90pRX90m, 8phaseL = 0, ampL = ampLX90p,

phaseR = 0, ampR = ampRX90p<D;

If@test1 ã LX90mRX90m, 8phaseL = 0, ampL = ampLX90m,

phaseR = 0, ampR = ampRX90m<D;

If@test1 ã LY90pRX90m, 8phaseL = p ê 2, ampL = ampLY90p,

phaseR = 0, ampR = ampRX90m<D;

If@test1 ã LY90mRX90m, 8phaseL = p ê 2, ampL = ampLY90m,

phaseR = 0, ampR = ampRX90m<D;

If@test1 ã LXpRX90m, 8phaseL = 0, ampL = ampLXp, phaseR = 0, ampR = ampRX90m<D;

phaseR = 0, ampR = ampRX90m<D;

If@test1 ã LXmRX90m, 8phaseL = 0, ampL = ampLXm, phaseR = 0, ampR = ampRX90m<D;

D;
If@test1 ã LYpRX90m,
262 mathematica code: pulse generation
8phaseL = p ê 2, ampL = ampLYp, phaseR = 0, ampR = ampRX90m<D;
If@test1 ã LYpRX90m,

If@test1 ã LYmRX90m, 8phaseL = p ê 2, ampL = ampLYm,

If@test1 ã LX90pRY90p, 8phaseL = 0, ampL = ampLX90p,

phaseR = 0, ampR = ampRX90m<D;

phaseR = p ê 2, ampR = ampRY90p<D;

If@test1 ã LX90mRY90p, 8phaseL = 0, ampL = ampLX90m,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LY90pRY90p, 8phaseL = p ê 2, ampL = ampLY90p,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LY90mRY90p, 8phaseL = p ê 2, ampL = ampLY90m,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LXpRY90p, 8phaseL = 0, ampL = ampLXp,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LXmRY90p, 8phaseL = 0, ampL = ampLXm,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LYpRY90p, 8phaseL = p ê 2, ampL = ampLYp,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LYmRY90p, 8phaseL = p ê 2, ampL = ampLYm,
phaseR = p ê 2, ampR = ampRY90p<D;
If@test1 ã LX90pRY90m, 8phaseL = 0, ampL = ampLX90p,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LX90mRY90m, 8phaseL = 0, ampL = ampLX90m,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LY90pRY90m, 8phaseL = p ê 2, ampL = ampLY90p,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LY90mRY90m, 8phaseL = p ê 2, ampL = ampLY90m,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LXpRY90m, 8phaseL = 0, ampL = ampLXp,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LXmRY90m, 8phaseL = 0, ampL = ampLXm,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LYpRY90m, 8phaseL = p ê 2, ampL = ampLYp,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LYmRY90m, 8phaseL = p ê 2, ampL = ampLYm,
phaseR = p ê 2, ampR = ampRY90m<D;
If@test1 ã LX90pRXp, 8phaseL = 0, ampL = ampLX90p, phaseR = 0, ampR = ampRXp<D;
If@test1 ã LX90mRXp, 8phaseL = 0, ampL = ampLX90m, phaseR = 0, ampR = ampRXp<D;

8phaseL = p ê 2, ampL = ampLY90p, phaseR = 0, ampR = ampRXp<D;

If@test1 ã LY90pRXp,

If@test1 ã LY90mRXp, 8phaseL = p ê 2, ampL = ampLY90m,

If@test1 ã LXpRXp, 8phaseL = 0, ampL = ampLXp, phaseR = 0, ampR = ampRXp<D;

phaseR = 0, ampR = ampRXp<D;

If@test1 ã LXmRXp, 8phaseL = 0, ampL = ampLXm, phaseR = 0, ampR = ampRXp<D;

If@test1 ã LYpRXp, 8phaseL = p ê 2, ampL = ampLYp, phaseR = 0, ampR = ampRXp<D;
If@test1 ã LYmRXp, 8phaseL = p ê 2, ampL = ampLYm, phaseR = 0, ampR = ampRXp<D;
If@test1 ã LX90pRXm, 8phaseL = 0, ampL = ampLX90p, phaseR = 0, ampR = ampRXm<D;
If@test1 ã LX90mRXm, 8phaseL = 0, ampL = ampLX90m, phaseR = 0, ampR = ampRXm<D;

8phaseL = p ê 2, ampL = ampLY90p, phaseR = 0, ampR = ampRXm<D;

If@test1 ã LY90pRXm,

If@test1 ã LY90mRXm, 8phaseL = p ê 2, ampL = ampLY90m,

If@test1 ã LXpRXm, 8phaseL = 0, ampL = ampLXp, phaseR = 0, ampR = ampRXm<D;

phaseR = 0, ampR = ampRXm<D;

If@test1 ã LXmRXm, 8phaseL = 0, ampL = ampLXm, phaseR = 0, ampR = ampRXm<D;

If@test1 ã LYpRXm, 8phaseL = p ê 2, ampL = ampLYp, phaseR = 0, ampR = ampRXm<D;
If@test1 ã LYmRXm, 8phaseL = p ê 2, ampL = ampLYm, phaseR = 0, ampR = ampRXm<D;

8phaseL = 0, ampL = ampLX90p, phaseR = p ê 2, ampR = ampRYp<D;

If@test1 ã LX90pRYp,

If@test1 ã LX90mRYp, 8phaseL = 0, ampL = ampLX90m,

phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã LY90pRYp, 8phaseL = p ê 2, ampL = ampLY90p,
phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã LY90mRYp, 8phaseL = p ê 2, ampL = ampLY90m,
, <D;
 If@test1 ã LY90mRYp, 8phaseL = p ê 2, ampL = ampLY90m,
a. mathematica code: pulse generation 263
phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã LXpRYp, 8phaseL = 0, ampL = ampLXp, phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã LXmRYp, 8phaseL = 0, ampL = ampLXm, phaseR = p ê 2, ampR = ampRYp<D;

8phaseL = p ê 2, ampL = ampLYp, phaseR = p ê 2, ampR = ampRYp<D;

If@test1 ã LYpRYp,

If@test1 ã LYmRYp, 8phaseL = p ê 2, ampL = ampLYm,

phaseR = p ê 2, ampR = ampRYp<D;
If@test1 ã LX90pRYm, 8phaseL = 0, ampL = ampLX90p,
phaseR = p ê 2, ampR = ampRYm<D;
If@test1 ã LX90mRYm, 8phaseL = 0, ampL = ampLX90m,
phaseR = p ê 2, ampR = ampRYm<D;
If@test1 ã LY90pRYm, 8phaseL = p ê 2, ampL = ampLY90p,
phaseR = p ê 2, ampR = ampRYm<D;
If@test1 ã LY90mRYm, 8phaseL = p ê 2, ampL = ampLY90m,
phaseR = p ê 2, ampR = ampRYm<D;
If@test1 ã LXpRYm, 8phaseL = 0, ampL = ampLXp, phaseR = p ê 2, ampR = ampRYm<D;
If@test1 ã LXmRYm, 8phaseL = 0, ampL = ampLXm, phaseR = p ê 2, ampR = ampRYm<D;

8phaseL = p ê 2, ampL = ampLYp, phaseR = p ê 2, ampR = ampRYm<D;

If@test1 ã LYpRYm,

If@test1 ã LYmRYm, 8phaseL = p ê 2, ampL = ampLYm,

phaseR = p ê 2, ampR = ampRYm<D;
H* Single Qubit Arbitrary Rotation Pulses with Arbitrary Sigma *L
If@test1 ã LXq, 8phaseL = 0, ampL = 2000, Ssigma = 2000;<D;
If@test1 ã LYq, 8phaseL = p ê 2, ampL = 2000, Ssigma = 2000<D;
If@test1 ã RXq, 8phaseR = 0, ampR = 2000, Ssigma = 2000<D;
If@test1 ã RYq, 8phaseR = p ê 2, ampR = 2000, Ssigma = 2000<D;

8phaseL = 0, ampL = 2000, phaseR = 0, ampR = 2000, Ssigma = 2000<D;

If@test1 ã LXqRXq,

If@test1 ã LXqRYq, 8phaseL = 0, ampL = 2000,

phaseR = p ê 2, ampR = 2000, Ssigma = 2000<D;
If@test1 ã LYqRXq, 8phaseL = p ê 2, ampL = 2000, phaseR = 0,

If@test1 ã LYqRYq, 8phaseL = p ê 2, ampL = 2000,

ampR = 2000, Ssigma = 2000<D;

phaseR = p ê 2, ampR = 2000, Ssigma = 2000<D;

H* Single Qubit Z-Pulses *L
If@test1 ã LZf, 8ampLZ = 2000, widthLZ = cycleLength, spec = 0, flux = 1<D;
If@test1 ã RZf, 8ampRZ = 2000, widthRZ = cycleLength, spec = 0, flux = 1<D;
If@test1 ã LZfRZf, 8ampLZ = 2000, ampRZ = 2000,

H* Two Qubit C-Phases Type a *L

widthLZ = cycleLength, widthRZ = cycleLength, spec = 0, flux = 1<D;

8ampLZ = ampC0L, ampRZ = ampC0R, widthLZ = fwidthL, widthRZ = fwidthR,

If@test1 ã C0,

If@test1 ã C1, 8ampLZ = ampC1L, ampRZ = ampC1R, widthLZ = fwidthL, widthRZ =

offsetLZ = offsetC0L, offsetRZ = offsetC0R, spec = 0, flux = 1<D;

If@test1 ã C2, 8ampLZ = ampC2L, ampRZ = ampC2R, widthLZ = fwidthL, widthRZ =

fwidthR, offsetLZ = offsetC1L, offsetRZ = offsetC1R, spec = 0, flux = 1<D;

If@test1 ã C3, 8ampLZ = ampC3L, ampRZ = ampC3R, widthLZ = fwidthL, widthRZ =

fwidthR, offsetLZ = offsetC2L, offsetRZ = offsetC2R, spec = 0, flux = 1<D;

H* Two Qubit C-Phases Type b, when following a C-Phase type a*L

fwidthR, offsetLZ = offsetC3L, offsetRZ = offsetC3R, spec = 0, flux = 1<D;

8ampLZ = ampC0Lb, ampRZ = ampC0Rb, widthLZ = fwidthL, widthRZ = fwidthR,

If@test1 ã C0b,

If@test1 ã C1b, 8ampLZ = ampC1Lb, ampRZ = ampC1Rb, widthLZ = fwidthL, widthRZ =

offsetLZ = offsetC0Lb, offsetRZ = offsetC0Rb, spec = 0, flux = 1<D;

If@test1 ã C2b, 8ampLZ = ampC2Lb, ampRZ = ampC2Rb, widthLZ = fwidthL, widthRZ =

fwidthR, offsetLZ = offsetC1Lb, offsetRZ = offsetC1Rb, spec = 0, flux = 1<D;

If@test1 ã C3b, 8ampLZ = ampC3Lb, ampRZ = ampC3Rb, widthLZ = fwidthL, widthRZ =

fwidthR, offsetLZ = offsetC2Lb, offsetRZ = offsetC2Rb, spec = 0, flux = 1<D;

8phaseL, ampL, phaseR, ampR, ampLZ, widthLZ, offsetLZ,

fwidthR, offsetLZ = offsetC3Lb, offsetRZ = offsetC3Rb, spec = 0, flux = 1<D;

ampRZ, widthRZ, offsetRZ, spec, flux, Ssigma<D;

ü Rabi angle to AWG DAC value function

264 mathematica code: pulse generation

Rabi angle to AWG DAC value function

AngletoDAC@Angle_, mPiDAC_, zeroDAC_D :=

HAngle + 180LF;
zeroDAC - mPiDAC
ModuleB8DACLevel<, DACLevel = RoundBmPiDAC +
180.
8DACLevel<F;

Pulse Functions

ü Various Gaussian and tangent hyperbolic spec pulse functions

DerivGaussianPulse@amp_, n_, s_D := ModuleB8midpoint<, midpoint = Hn + 1L ê 2;

Hx - midpointL Hx - midpointL2
Round BTableB- amp ExpB- F, 8x, 1, n<FFF
s2 2 s2

GaussianPulseSubOff@amp_, n_, s_D := ModuleB8midpoint<, midpoint = Hn + 1L ê 2;

Ix-midpointM
F - ExpB F
2 2
-ImidpointM
ExpB-
Round BTableBamp , 8x, 1, n<FFF
2 s2 2 s2

F
2
-ImidpointM
1 - ExpB
2 s2

DerivGaussianPulseSubOff@amp_, n_, s_D :=

ModuleB8midpoint<, midpoint = Hn + 1L ê 2;
Hx - midpointL
Round BTableB- amp
s2
Ix-midpointM
F - ExpB F
2 2
-ImidpointM
ExpB-
, 8x, 1, n<FFF
2 s2 2 s2

F
2
-ImidpointM
1 - ExpB
2 s2

SquareGaussPow@amp_, n_, m_, s_, p_D := ModuleB8t0, t1<,

H* amp is the max amplitude,
n is the total length, m is the number of sigmas for truncation,

t0 = Hm s + 1L ê 2;
s is the std. dev, p is the root of gaussian to take*L

t1 = n - Hm s + 1L ê 2;
RoundBTableB amp ê 2 HeavisideTheta@t0 - tD

Ht - t0L2
F + amp HHHeavisideTheta@t - t0D + HeavisideTheta@t1 - tDL ê 2L +
1ëp

ExpB-
2 s2

Ht - t1L2
amp ê 2 HeavisideTheta@t - t1D ExpB- F , 8t, 1, n<FFF
1ëp

2 s2

ü
 a. mathematica code: pulse generation 265
TanhPulse@amp_, s_, length_D := Module@8t0, t1<,
t0 = 4 s;
t1 = 4 s + length;

amp HTanh@Ht - t0L ê sD + Tanh@-Ht - t1L ê sDL ê 2, 8t, 0, Round@length + 8 sD<DD

Round@Table@

D;

TanhPulsewOffset@amp_, s_, length_, offset_D := Module@8t0, t1<,

t0 = 4 s;

Round@Table@amp HTanh@Ht - t0L ê sD + Tanh@-Ht - t1L ê sDL ê 2 +

t1 = 4 s + length;

offset * HKroneckerDelta@t, t0D + KroneckerDelta@t, t1DL ,

8t, 0, Round@length + 8 sD<DD
D;

trueoffset = 8Round@offset ê 100 * amp * Exp@-HnumSigmas ê 2L ^ 2 ê 2DD<;

GaussPulsewOffset@amp_, numSigmas_, length_, offset_D := Module@8trueoffset<,

GaussianPulse@amp, Hlength - 2L, Hlength - 2L ê numSigmasD, trueoffsetDD;

Join@trueoffset,
266 mathematica code: pulse generation

ü Building up spectroscopy and flux pulses

H* Flux pulse with rounded top *L
FluxPulse@amp_, numSigmas_, length_,
offset_, fixedPoint_, cycleLength_, FluxDelay_D :=

fixedPoint + FluxDelay, 8<, cycleLengthD;

MakePattern@GaussPulsewOffset@amp, numSigmas, length, offsetD,

H* DPS scale factor D *L

DerivAmpScaleL = 0.5;

H* Spectroscopy pulse with option for using DPS,

DerivAmpScaleR = 0.2;

gaussian, square, or tanh *L

FullSpecPulse@amp_, SpecPulseLength_,
fixedPoint_, cycleLength_, SpecDelay_, type_D :=
MakePattern@Which@type ã 0 , GaussianPulseSubOff@amp,
numSigmas * SpecPulseLength, SpecPulseLengthD,
type ã 1, SquarePulse@amp,
numSigmas * SpecPulseLengthD,
type ã 2, TanhPulse@amp, tanhSigma,
numSigmas * SpecPulseLength - 8 * tanhSigmaD,
type ã 3, InputPulse@amp,
SpecPulseLength, SpecPulseLength, SpecPointsID,
type ã 4, InputPulse@amp,
SpecPulseLength, SpecPulseLength, SpecPointsQD,
type ã 6, DerivGaussianPulseSubOff@

D, fixedPoint + SpecDelay, 8<, cycleLengthD;

amp, numSigmas * SpecPulseLength, SpecPulseLengthD

H* Single spectroscopy pulse, used for Rabi,

T1 and as bulding block for Ramsey *L
SpecPulse@amp_, SpecPulseLength_, fixedPoint_, cycleLength_, SpecDelay_D :=

fixedPoint + SpecDelay, 8<, cycleLengthD;

MakePattern@GaussianPulse@amp, numSigmas * SpecPulseLength, SpecPulseLengthD,

H* Measurement Pulse *L
MeasPulse@MeasPulseLength_, fixedPoint_, cycleLength_D :=
MakePattern@8<, fixedPoint, SquarePulse@1, MeasPulseLengthD, cycleLengthD;

Join üü Table@MeasPulse@MeasPulseLength, fixedPoint, cycleLengthD, 8n<D;

MeasPulseArray@n_, MeasPulseLength_, fixedPoint_, cycleLength_D :=

H* Blank out Pulse *L

BlankPulse@BlankPulseLength_, fixedPoint_, cycleLength_D :=

MakePattern@8<, fixedPoint, SquarePulse@0, BlankPulseLengthD, cycleLengthD;

Join üü Table@BlankPulse@BlankPulseLength, fixedPoint, cycleLengthD, 8n<D;

BlankPulseArray@n_, BlankPulseLength_, fixedPoint_, cycleLength_D :=

H* Buffer pulse *L
SpecBufferPulse@specBufferLength_, endPoint_, cycleLength_,

endPoint + SpecBufferDelay, 8<, cycleLengthD;

SpecBufferDelay_D := MakePattern@SquarePulse@1, specBufferLengthD,

Pattern Functions
The following function parses through sequences of many pulses, comprised of both qubit driving pulses and
flux pulses. It calls PulseIdentify[-] to assign the appropriate times, delays, amplitudes, and offsets.
GetNewSeq@seq_, SpecAmpL_, SpecAmpR_, numSigmasfL_, numSigmasfR_,
fampL_, fampR_, widthfL_, widthfR_, specType_, sigma_, awgType_D :=
Module@8seqLX, seqLY, aL, L, seqRX, seqRY, aR, R, seqLZ, aLZ, wLZ,
oLZ, seqRZ, aRZ, wRZ, oRZ, s, f, widthF, widthSL, widthSR, bufferL,
bufferR, Ssigma, typeI, typeQ, typeIpQ, input, quadspec,
cLength, fPoint, AWGDelay, sigma7102, bufferSL, bufferSR<,
 a. mathematica code: pulse generation
bufferR, Ssigma, typeI, typeQ, typeIpQ, input, quadspec,
267
cLength, fPoint, AWGDelay, sigma7102, bufferSL, bufferSR<,

H* Default values *L
seqLX = 0; seqLY = 0; seqRX = 0; seqRY = 0; seqLZ = 0; seqRZ = 0;
aL = DACoffsetLX;
aR = DACoffsetRX;
widthF = 0; widthSL = 0; widthSR = 0; bufferL = 0; bufferR = 0;
input = 0;
quadspec = 0;
cLength = cycleLength;
fPoint = fixedPointCh1;
AWGDelay = AWG520Delay;

For@j = Length@seqD, j ¥ 1, j --, 8

bufferSL = bufferS; bufferSR = bufferS;

8L, aL, R, aR, aLZ, wLZ, oLZ, aRZ, wRZ, oRZ, s, f, Ssigma< =
PulseIdentify@seq@@jDDD ;
typeI = specType;
typeQ = specType;

If@specType ã 6, 8quadspec = 1, typeI = 0, typeQ = 0, typeIpQ = 6<D;

typeIpQ = 0;

8typeI = 3, typeQ = 4, input = 1, s = 0, aL = SpecAmpL, L = 0, Ssigma = sigma<D;

If@seq@@jDD ã LXi,

If@seq@@jDD ã LYi, 8typeI = 4, typeQ = 3, input = 1,

If@seq@@jDD ã RXi, 8typeI = 3, typeQ = 4, input = 1, s = 0,

s = 0, aL = SpecAmpL, L = 0, Ssigma = sigma<D;

If@seq@@jDD ã RYi, 8typeI = 4, typeQ = 3, input = 1, s = 0,

aR = SpecAmpR, R = 0, Ssigma = sigma<D;

H* For arbitrary flux amplitude pulses *L

aR = SpecAmpR, R = 0, Ssigma = sigma<D;

If@aL ã 2000, 8aL = SpecAmpL<D;

If@aR ã 2000, 8aR = SpecAmpR<D;
If@aLZ ã 2000, aLZ = fampLD;
If@aRZ ã 2000, aRZ = fampRD;
If@wLZ ã cycleLength, wLZ = widthfLD;
If@wRZ == cycleLength, wRZ = widthfRD;
If@Ssigma ã 2000, Ssigma = sigmaD;

seqLX += Cos@LD FullSpecPulse@aL - DACoffsetLX, Ssigma,

fPoint - bufferL - widthF - widthSL, cLength, SpecPulseDelay, typeID +
input * Sin@LD FullSpecPulse@aL - DACoffsetLX, Ssigma,

H*quadspec*Sin@LD FullSpecPulse@aL-DACoffsetLX,Ssigma,
fPoint - bufferL - widthF - widthSL, cLength, SpecPulseDelay, typeID +

quadspec * Sin@LD FullSpecPulse@DerivAmpScaleL HaL - DACoffsetLXL,

fPoint-bufferL-widthF-widthSL,cLength,SpecPulseDelay,typeIpQD;*L

sigma7102, fPoint - bufferL - widthF - widthSL,

cLength, SpecPulseDelay, 6D;
seqLY += Sin@LD FullSpecPulse@aL - DACoffsetLY, Ssigma,
fPoint - bufferL - widthF - widthSL, cLength, SpecPulseDelay, typeQD +
input * Cos@LD FullSpecPulse@aL - DACoffsetLY, Ssigma,

H*quadspec*Cos@LD FullSpecPulse@aL-DACoffsetLX,Ssigma,
fPoint - bufferL - widthF - widthSL, cLength, SpecPulseDelay, typeQD -

quadspec * Cos@LD FullSpecPulse@DerivAmpScaleL HaL - DACoffsetLXL,

fPoint-bufferL-widthF-widthSL,cLength,SpecPulseDelay,typeIpQD;*L

sigma7102, fPoint - bufferL - widthF - widthSL,

cLength, SpecPulseDelay, 6D;
seqRX += Cos@RD FullSpecPulse@aR - DACoffsetRX, Ssigma,
fixedPointCh1 - bufferR - widthF - widthSR, cycleLength,
SpecPulseDelay + AWGDelay, typeID + input * Sin@RD
FullSpecPulse@aR - DACoffsetLX, Ssigma, fixedPointCh1 - bufferR -

quadspec * Sin@RD FullSpecPulse@DerivAmpScaleR HaR - DACoffsetLXL,

widthF - widthSR, cycleLength, SpecPulseDelay + AWGDelay, typeID +

Ssigma, fixedPointCh1 - bufferR - widthF - widthSR,

cycleLength, , 6D ;
268 mathematica code: pulse generation
Ssigma, fixedPointCh1 - bufferR - widthF - widthSR,
cycleLength, SpecPulseDelay + AWGDelay, 6D ;
seqRY += Sin@RD FullSpecPulse@aR - DACoffsetRY, Ssigma,
fixedPointCh1 - bufferR - widthF - widthSR, cycleLength,
SpecPulseDelay + AWGDelay, typeQD + input * Cos@RD
FullSpecPulse@aR - DACoffsetLY, Ssigma, fixedPointCh1 - bufferR -

quadspec * Cos@RD FullSpecPulse@DerivAmpScaleR HaR - DACoffsetLXL,

widthF - widthSR, cycleLength, SpecPulseDelay + AWGDelay, typeQD -

Ssigma, fixedPointCh1 - bufferR - widthF - widthSR,

cycleLength, SpecPulseDelay + AWGDelay, 6D ;
seqLZ += FluxPulse@aLZ - DACoffsetLZ, numSigmasfL, wLZ, oLZ,
fixedPointCh1 - bufferR - widthF - widthSR, cycleLength, FluxPulseDelayD;
seqRZ += FluxPulse@aRZ - DACoffsetLZ, numSigmasfL, wRZ, oRZ,
fixedPointCh1 - bufferR - widthF - widthSR, cycleLength, FluxPulseDelayD;

widthF += Max@wLZ, wRZD * f;

widthSL += numSigmas * Ssigma * s + Length@SpecPointsID * input;

bufferL += bufferSL * Hinput + sL + bufferF * f;

widthSR += numSigmas * Ssigma * s + Length@SpecPointsID * input;

bufferR += bufferSL * Hinput + sL + bufferF * f;

<D;
Round êü 8Clip@seqLX + DACoffsetLX, 80, 1023<D,
Clip@seqLY + DACoffsetLY, 80, 1023<D, Clip@seqRX + DACoffsetRX, 80, 1023<D,
Clip@seqRY + DACoffsetRY, 80, 1023<D,
Clip@seqLZ + DACoffsetLZ, 80, 1023<D, Clip@seqRZ + DACoffsetRZ, 80, 1023<D,
widthF, widthSL, widthSR, bufferL, bufferR<D;

Calibration Sequences
The calibration sequences here refer to tuning up the DAC levels for p and p/2 pulses based on Rabi driving,
the drive detuning based off of Ramsey fringe experiments, the level of DPS to use based on the ALLXY
sequences consisting of pairs of X and Y pulses, as well as the conditional phase flux pulse amplitudes and
offsets. All of these routines involve:
1. Defining the pulse sequences
2. Generating the pattern files for each AWG channel
3. Generating the marker pattern files for measurement, spectroscopy buffering, and acquisition card triggering
4. Exporting the files

Each section below will only give the steps 1 and 2, defining the pulse sequences and pattern files. Subse-
quently, steps 3 to 4 are the same for all cases and are given at the end.
 a. mathematica code: pulse generation 269

Amplitude Rabi

ü Define pulse sequences and file directories

basename = "ARabi";
datapathAWG5014 = RootAWG5014 <> basename <> "_5014\\";
datapathAWG520 = RootAWG520 <> basename <> "_520\\";
CreateDirectory@datapathAWG5014D;
CreateDirectory@datapathAWG520D;
filePrefix5014 = basename <> "_5014";
filePrefix520 = basename <> "_520";

ExpSeq = Table@8LXqRXq<, 8k, 1, numstepsExp<D;

numstepsExp = 80;

numsteps = Length@ExpSeqD;

ü Generate pulse patterns

H* Initialization *L

patTableCh2 = patTableCh3 = patTableCh4 = patTableCh5 = patTableCh6 = 8<;

patTableCh1 =

SpecBufferWidthPointsL = 8<; SpecBufferWidthPointsR = 8<;

RabiCenter = 0; RabiStep = 17;

Table@RabiCenter + Hii - numstepsExp ê 2L * RabiStep, 8ii, 0, numsteps - 1<D;

RabiPointsL = DACoffsetLX +

RabiPointsR = DACoffsetRX + Table@RabiCenter + Hii - numstepsExp ê 2L * RabiStep,

8ii, 0, numsteps - 1<D;

Monitor@Do@
ampL = RabiPointsL@@iiDD; ampR = RabiPointsR@@iiDD;
numSigmasFL = 0.000001; numSigmasFR = 0.000001;
widthFL = 0; widthFR = 0;

8patLX, patLY, patRX, patRY, patLZ,

ampFL = DACoffsetLZ; ampFR = DACoffsetRZ;

patRZ, widthF, widthSL, widthSR, bufferL, bufferR< =

GetNewSeq@ExpSeq@@iiDD, RabiPointsL@@iiDD, RabiPointsR@@iiDD, numSigmasFL,
numSigmasFR, ampFL, ampFR, widthFL, widthFR, 0, SpecPulseLength, 7D;
patTableCh1 = Append@patTableCh1, patLXD;
patTableCh2 = Append@patTableCh2, patLYD;
patTableCh3 = Append@patTableCh3, patLZD;
patTableCh4 = Append@patTableCh4, patRZD;
patTableCh5 = Append@patTableCh5, patRXD;
patTableCh6 = Append@patTableCh6, patRYD;
totalTimeL = widthSL + widthF + bufferL;
totalTimeR = widthSR + widthF + bufferR;
SpecBufferWidthPointsL =
Append@SpecBufferWidthPointsL, totalTimeL + 2 * SpecBufferMargin7102D;
SpecBufferWidthPointsR = Append@SpecBufferWidthPointsR,

, 8ii, 1, numsteps<D, iiD;

totalTimeR + 2 * SpecBufferMarginD;
270 mathematica code: pulse generation

Ramsey

ü Define pulse sequences

basename = "Ram";

ExpSeq = Table@8LX90pRX90p, LZfRZf, LX90pRX90p<, 8k, 1, numstepsExp<D;

numstepsExp = 80;

numsteps = Length@ExpSeqD;

ü Generate pulse patterns

H* Initialization *L

patTableCh2 = patTableCh3 = patTableCh4 = patTableCh5 = patTableCh6 = 8<;

patTableCh1 =

SpecBufferWidthPointsL = 8<; SpecBufferWidthPointsR = 8<;

RamseyStart = 0; RamseyStepR = 20; RamseyStepL = 200; H* in ns *L
RamseyPointsR = Table@RamseyStart + ii RamseyStepR, 8ii, 0, numsteps - 1<D;
RamseyPointsL = Table@RamseyStart + ii RamseyStepL, 8ii, 0, numsteps - 1<D;
Monitor@Do@
ampL = DACoffsetLX; ampR = DACoffsetRX;
numSigmasFL = 0.000001; numSigmasFR = 0.000001;
widthFL = RamseyPointsL@@iiDD; widthFR = RamseyPointsR@@iiDD;

8patLX, patLY, patRX, patRY, patLZ, patRZ, widthF, widthSL, widthSR,

ampFL = DACoffsetLZ; ampFR = DACoffsetRZ;

bufferL, bufferR< = GetNewSeq@ExpSeq@@iiDD, ampL, ampR, numSigmasFL,

numSigmasFR, ampFL, ampFR, widthFL, widthFR, 11, SpecPulseLength, 7D;
patTableCh1 = Append@patTableCh1, patLXD;
patTableCh2 = Append@patTableCh2, patLYD;
patTableCh3 = Append@patTableCh3, patLZD;
patTableCh4 = Append@patTableCh4, patRZD;
patTableCh5 = Append@patTableCh5, patRXD;
patTableCh6 = Append@patTableCh6, patRYD;
totalTimeL = widthSL + widthF + bufferL;
totalTimeR = widthSR + widthF + bufferR;
SpecBufferWidthPointsL =
Append@SpecBufferWidthPointsL, totalTimeL + 2 * SpecBufferMargin7102D;
SpecBufferWidthPointsR = Append@SpecBufferWidthPointsR,

, 8ii, 1, numsteps<D, iiD;

totalTimeR + 2 * SpecBufferMarginD;

All XY

ü Define pulse sequences

runType = 0;
basename = Which@runType ã 0, "ALLXY_L", runType == 1, "AllXY_R"D;
 a. mathematica code: pulse generation 271
ExpSeq = Which@runType ã 0,

88Id<, 8Id<, 8LXp<, 8LXp<, 8LYp<, 8LYp<, 8LX90p<, 8LX90p<, 8LY90p<, 8LY90p<,
8LXp, LXp<, 8LXp, LXp<, 8LXp, LYp<, 8LXp, LYp<,
8LYp, LXp<, 8LYp, LXp<, 8LYp, LYp<, 8LYp, LYp<,
8LXp, LX90p<, 8LXp, LX90p<, 8LXp, LY90p<, 8LXp, LY90p<, 8LYp, LX90p<,
8LYp, LX90p<, 8LYp, LY90p<, 8LYp, LY90p<, 8LX90p, LXp<, 8LX90p, LXp<,
8LX90p, LYp<, 8LX90p, LYp<, 8LY90p, LXp<, 8LY90p, LXp<,
8LY90p, LYp<, 8LY90p, LYp<, 8LX90p, LX90p<, 8LX90p, LX90p<,
8LX90p, LY90p<, 8LX90p, LY90p<, 8LY90p, LX90p<,
8LY90p, LX90p<, 8LY90p, LY90p<, 8LY90p, LY90p<
<,

88Id<, 8Id<, 8RXp<, 8RXp<, 8RYp<, 8RYp<, 8RX90p<, 8RX90p<,

runType ã 1,

8RY90p<, 8RY90p<, 8RXp, RXp<, 8RXp, RXp<, 8RXp, RYp<, 8RXp, RYp<,
8RYp, RXp<, 8RYp, RXp<, 8RYp, RYp<, 8RYp, RYp<, 8RXp, RX90p<,
8RXp, RX90p<, 8RXp, RY90p<, 8RXp, RY90p<, 8RYp, RX90p<, 8RYp, RX90p<,
8RYp, RY90p<, 8RYp, RY90p<, 8RX90p, RXp<, 8RX90p, RXp<, 8RX90p, RYp<,
8RX90p, RYp<, 8RY90p, RXp<, 8RY90p, RXp<, 8RY90p, RYp<, 8RY90p, RYp<,
8RX90p, RX90p<, 8RX90p, RX90p<, 8RX90p, RY90p<, 8RX90p, RY90p<,
8RY90p, RX90p<, 8RY90p, RX90p<, 8RY90p, RY90p<, 8RY90p, RY90p<
<
D;
numsteps = Length@ExpSeqD

ü Generate pulse patterns

H* Initialization *L

patTableCh2 = patTableCh3 = patTableCh4 = patTableCh5 = patTableCh6 = 8<;

patTableCh1 =

SpecBufferWidthPointsL = 8<;
SpecBufferWidthPointsR = 8<;
RabiCenter = 0; RabiStep = 16;

Table@RabiCenter + Hii - numsteps1 ê 2L * RabiStep, 8ii, 0, numsteps - 1<D;

RabiPointsL = DACoffsetLX +

RabiPointsR = RabiPointsL;
Monitor@Do@
ampL = RabiPointsL@@iiDD;
ampR = RabiPointsR@@iiDD;
numSigmasFL = 0.000001;
numSigmasFR = 0.000001;
widthFL = 0;
widthFR = 0;
ampFL = DACoffsetLZ;

8patLX, patLY, patRX, patRY, patLZ, patRZ, widthF, widthSL,

ampFR = DACoffsetRZ;

widthSR, bufferL, bufferR< = GetNewSeq@ExpSeq@@iiDD, ampL, ampR,

numSigmasFL, numSigmasFR, ampFL, ampFR, widthFL, widthFR, 11, 1, 7D;
patTableCh1 = Append@patTableCh1, patLXD;
patTableCh2 = Append@patTableCh2, patLYD;
patTableCh3 = Append@patTableCh3, patLZD;
patTableCh4 = Append@patTableCh4, patRZD;
patTableCh5 = Append@patTableCh5, patRXD;
patTableCh6 = Append@patTableCh6, patRYD;
totalTimeL = widthSL + widthF + bufferL;
totalTimeR = widthSR + widthF + bufferR;
SpecBufferWidthPointsL =
Append@SpecBufferWidthPointsL, totalTimeL + 2 * SpecBufferMarginD;
SpecBufferWidthPointsR = Append@SpecBufferWidthPointsR,

, 8ii, 1, numsteps<D, iiD;

totalTimeR + 2 * SpecBufferMarginD;

Conditional Phase Tune-Up

272 mathematica code: pulse generation

Conditional Phase Tune-Up

ü Define pulse sequences

runType = 2; H* 0 instructs to sweep pulse width,
1 instructs to sweep left dynamic phase,
2 instrucs to sweep right dynamic phase *L
basename = Which@runType ã 0, "CPHL_vW",

SeqEx1 = Table@ Which@runType ã 0, 8LY90p, LZfRZf, LY90p<,

runType ã 1, "CPHL_vDL", runType ã 2, "CPHR_vDR"D

runType ã 1, 8LY90p, LZfRZf, LY90p<,

runType ã 2, 8RY90p, LZfRZf, RY90p<D, 8k, 1, 101<D;
SeqEx2 = Table@ Which@runType ã 0, 8LY90pRYp, LZfRZf , LY90pRYp<,
runType ã 1, 8LY90pRYp, LZfRZf, LY90pRYp<,

runType ã 2, 8LYpRY90p, LZfRZf, LYpRY90p<D, 8k, 1, 101<D;

IdSeq = Table@8Id<, 8k, 1, 5<D;
LXpSeq = Table@8LXp<, 8k, 1, 5<D;
RXpSeq = Table@8RXp<, 8k, 1, 5<D;
LXpRXpSeq = Table@8LXpRXp<, 8k, 1, 5<D;
SeqEx1 = Join@SeqEx1, IdSeq, LXpSeq, RXpSeq, LXpRXpSeqD;
SeqEx2 = Join@SeqEx2, IdSeq, LXpSeq, RXpSeq, LXpRXpSeqD;
numsteps = Length@SeqEx1D
 a. mathematica code: pulse generation 273

ü Generate pattern files

H* Initialization *L
patTableCh1Ex1 = 8<; patTableCh2Ex1 = 8<; patTableCh3Ex1 = 8<;
patTableCh4Ex1 = 8<; patTableCh5Ex1 = 8<; patTableCh6Ex1 = 8<;
patTableCh1Ex2 = 8<; patTableCh2Ex2 = 8<; patTableCh3Ex2 = 8<;
patTableCh4Ex2 = 8<; patTableCh5Ex2 = 8<; patTableCh6Ex2 = 8<;
SpecBufferWidthPointsEx1 = 8<; SpecBufferWidthPointsEx2 = 8<;

FluxPulseAmpLStart = Which@ runType ã 0,

DACoffsetLZ, runType ã 1, DACoffsetLZ , runType ã 2, 660D;
FluxPulseAmpLStep = Which@ runType ã 0, 0,
runType ã 1, 10, runType ã 2, 0D;
FluxPulseAmpRStart = Which@ runType ã 0,
0, runType ã 1, 1000, runType ã 2, 1000D;
FluxPulseAmpRStep = Which@ runType ã 0,
0, runType ã 1, 0, runType ã 2, 0D;
FluxPulseOffsetLStart = Which@ runType ã 0,
0, runType ã 1, 0, runType ã 2, 0D;
FluxPulseOffsetLStep = Which@ runType ã 0,
0, runType ã 1, 0, runType ã 2, 0D;
FluxPulseOffsetRStart = Which@ runType ã 0,
0, runType ã 1, 0, runType ã 2, 0D;
FluxPulseOffsetRStep = Which@ runType ã 0,
0, runType ã 1, 0, runType ã 2, 1D;
FluxPulseWidthStart = Which@ runType ã 0, 0,
runType ã 1, 30, runType ã 2, 30D;
FluxPulseWidthStep = Which@ runType ã 0, 1,
runType ã 1, 0, runType ã 2, 0D;

8ii, 0, numsteps - 1<D;

FluxPulseOffsetLPoints = Table@FluxPulseOffsetLStart + ii FluxPulseOffsetLStep,

8ii, 0, numsteps - 1<D;

FluxPulseOffsetRPoints = Table@FluxPulseOffsetRStart + ii FluxPulseOffsetRStep,

8ii, 0, numsteps - 1<D;

FluxPulseAmpLPoints = Table@FluxPulseAmpLStart + ii FluxPulseAmpLStep,

8ii, 0, numsteps - 1<D;

FluxPulseAmpRPoints = Table@FluxPulseAmpRStart + ii FluxPulseAmpRStep,

8ii, 0, numsteps - 1<D;

FluxPulseWidthPoints = Table@FluxPulseWidthStart + ii FluxPulseWidthStep,

numSigmasFL = 0.000001;
numSigmasFR = 0.000001;

ampFL = 8FluxPulseAmpLPoints@@iiDD, FluxPulseAmpLPoints@@iiDD<;

Monitor@Do@

ampFR = 8FluxPulseAmpRPoints@@iiDD, FluxPulseAmpRPoints@@iiDD<;

offsetFL = 8FluxPulseOffsetLPoints@@iiDD, FluxPulseOffsetLPoints@@iiDD<;
offsetFR = 8FluxPulseOffsetRPoints@@iiDD, FluxPulseOffsetRPoints@@iiDD<;
widthFL = 8FluxPulseWidthPoints@@iiDD, FluxPulseWidthPoints@@iiDD<;
widthFR = 8FluxPulseWidthPoints@@iiDD, FluxPulseWidthPoints@@iiDD<;

8patLX, patLY, patRX, patRY, patLZ, patRZ, widthF, currdelay, widthS< =

GetSeqXYZFluxGauss@SeqEx1@@iiDD, ampFL, ampFR, widthFL,
widthFR, offsetFL, offsetFR, numSigmasFL, numSigmasFRD;
patTableCh1Ex1 = Append@patTableCh1Ex1, patLXD;
patTableCh2Ex1 = Append@patTableCh2Ex1, patLYD;
patTableCh3Ex1 = Append@patTableCh3Ex1, patLZD;
patTableCh4Ex1 = Append@patTableCh4Ex1, patRZD;
patTableCh5Ex1 = Append@patTableCh5Ex1, patRXD;
patTableCh6Ex1 = Append@patTableCh6Ex1, patRYD;
totalTime = widthS + widthF + currdelay;
SpecBufferWidthPointsEx1 =
;
274 mathematica code: pulse generation
SpecBufferWidthPointsEx1 =

, 8ii, 1, numsteps<D, iiD;

Append@SpecBufferWidthPointsEx1, totalTime + 2 * SpecBufferMarginD;

ampFL = 8FluxPulseAmpLPoints@@iiDD, FluxPulseAmpLPoints@@iiDD<;

Monitor@Do@

ampFR = 8FluxPulseAmpRPoints@@iiDD, FluxPulseAmpRPoints@@iiDD<;

offsetFL = 8FluxPulseOffsetLPoints@@iiDD, FluxPulseOffsetLPoints@@iiDD<;
offsetFR = 8FluxPulseOffsetRPoints@@iiDD, FluxPulseOffsetRPoints@@iiDD<;
widthFL = 8FluxPulseWidthPoints@@iiDD, FluxPulseWidthPoints@@iiDD<;
widthFR = 8FluxPulseWidthPoints@@iiDD, FluxPulseWidthPoints@@iiDD<;

8patLX, patLY, patRX, patRY, patLZ, patRZ, widthF, currdelay, widthS< =

GetSeqXYZFluxGauss@SeqEx2@@iiDD, ampFL, ampFR, widthFL,
widthFR, offsetFL, offsetFR, numSigmasFL, numSigmasFRD;
patTableCh1Ex2 = Append@patTableCh1Ex2, patLXD;
patTableCh2Ex2 = Append@patTableCh2Ex2, patLYD;
patTableCh3Ex2 = Append@patTableCh3Ex2, patLZD;
patTableCh4Ex2 = Append@patTableCh4Ex2, patRZD;
patTableCh5Ex2 = Append@patTableCh5Ex2, patRXD;
patTableCh6Ex2 = Append@patTableCh6Ex2, patRYD;
totalTime = widthS + widthF + currdelay;
SpecBufferWidthPointsEx2 =

, 8ii, 1, numsteps<D, iiD;

Append@SpecBufferWidthPointsEx2, totalTime + 2 * SpecBufferMarginD;

H* Join the two experiments *L

Table@Join@patTableCh1Ex1@@iiDD, patTableCh1Ex2@@iiDDD, 8ii, numsteps<D;

patTableCh1 =

8ii, numsteps<D;
patTableCh2 = Table@Join@patTableCh2Ex1@@iiDD, patTableCh2Ex2@@iiDDD,

8ii, numsteps<D;
patTableCh3 = Table@Join@patTableCh3Ex1@@iiDD, patTableCh3Ex2@@iiDDD,

8ii, numsteps<D;
patTableCh4 = Table@Join@patTableCh4Ex1@@iiDD, patTableCh4Ex2@@iiDDD,

8ii, numsteps<D;
patTableCh5 = Table@Join@patTableCh5Ex1@@iiDD, patTableCh5Ex2@@iiDDD,

8ii, numsteps<D;
patTableCh6 = Table@Join@patTableCh6Ex1@@iiDD, patTableCh6Ex2@@iiDDD,

Define AWG marker patterns

 a. mathematica code: pulse generation 275
H*AWG5014 Qubit 1 Markers*L
markerTable1 = Table@

8ii, numsteps<D;H* Marker 1 is a trigger *L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, cycleLengthD,

markerTable2 = Table@

SpecBufferMargin, cycleLength, SpecBufferDelayD, 8ii, numsteps<D;

SpecBufferPulse@SpecBufferWidthPointsR@@iiDD, fixedPointCh1 +

H* Marker 2 is the spec buffer for Qubit 1 Channels *L

markerTable3 = Table@

8ii, numsteps<D;H* Marker 3 is a trigger*L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, 1 * cycleLengthD,

markerTable4 = markerTable2;H* Marker 4 is the spec

H*AWG5014 FBL Markers*L

buffer for Qubit 1 Channels*L

markerTable5 = Table@

8ii, numsteps<D;H* Marker 5 is a trigger*L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, 1 * cycleLengthD,

markerTable6 = Table@

cycleLengthD, 8ii, numsteps<D;H* Marker 6 i a measurement pulse*L

MeasPulseArray@1, MeasPulseLength, fixedPointCh1 + MeasPulseDelay,

markerTable7 = Table@

8ii, numsteps<D;H* Marker 7 is a trigger*L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, 1 * cycleLengthD,

markerTable8 = Table@

fixedPointCh1 + MeasPulseDelay, cycleLengthD, 8ii, numsteps<D;

MeasPulseArray@1, MeasPulseLength,

H* Marker 8 is a measurement pulse*L

H*AWG520 QR Markers*L
markerTable9 = Table@

8ii, numsteps<D;H* Marker 9 is a trigger*L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, 1 * cycleLengthD,

markerTable10 = Table@
SpecBufferPulse@SpecBufferWidthPointsR@@iiDD,

SpecBufferDelay + AWG520DelayD, 8ii, numsteps<D;

fixedPointCh1 + SpecBufferMargin, cycleLength,

H* Marker 10 is the spec buffer for Qubit 2 Channels*L

markerTable11 = Table@

8ii, numsteps<D;H* Marker 11 is a trigger*L

PadRight@Join@Delay@1D, SquarePulse@1, triggerWidthDD, 1 * cycleLengthD,

markerTable12 = Table@

fixedPointCh1 + MeasPulseDelay + AWG520Delay, cycleLengthD, 8ii, numsteps<D;

MeasPulseArray@1, MeasPulseLength,

H* Marker 12 is a measurement pulse*L

Export pulse pattern files

H* Export pulses *L
steps = Range@numstepsD;
ExportTekPatternsAWG5014@datapathAWG5014, filePrefix5014,
patTableCh1, markerTable1, markerTable2, patTableCh2, markerTable3,
markerTable4, patTableCh3, markerTable5, markerTable6,
patTableCh4, markerTable7, markerTable8, steps, parameterListD
ExportTekPatternsAWG520@datapathAWG520, filePrefix520,
patTableCh5, markerTable9, markerTable10, patTableCh6,
markerTable11, markerTable12, steps, parameterListD
276 mathematica code: pulse generation

1-Qubit Benchmarking
Pulse sequences and functions for single-qubit randomized benchmarking (pattern generation, marker genera-
tion, and file exporting omitted)

ü Tools
Pauli and the Clifford generators

Pauli = 88Idp, Idm<, 8Xp, Xm<, 8Yp, Ym<, 8Zp, Zm<<;

Clifford = 88X90p, X90m<, 8Y90p, Y90m<<;
Undo = 88Idp, Idm<, 8X90p, X90m<, 8Y90p, Y90m<<;

Getting the generating function

signc, signp, undo, undotemp, zswap, signtemp<, gates = 8<;

Gates@NumberOfGates_D := Module@8gates, cgate, pgate,

cgate = CallClifford@@1DD;
pgate = CallPauli@@1DD;

If@pgate == 4, 8zswap = -zswap, pgate = 1<D;

zswap = 1;

signtemp = CallSignC@@1DD;
signc = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
signtemp = CallSignP@@1DD;
signp = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
undotemp = cgate;
undo = undotemp;
gates = Append@gates, Pauli@@pgate, signpDDD;

For@i = 1, i < NumberOfGates, 8

gates = Append@gates, Clifford@@cgate, signcDDD;

cgate = CallClifford@@iDD;

If@pgate == 4, 8zswap = -zswap, pgate = 1<D;

pgate = CallPauli@@iDD;

signtemp = CallSignC@@iDD;
signc = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
signtemp = CallSignP@@iDD;
signp = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
If@undo ã 0, undotemp = cgateD;
If@undo ã cgate, undotemp = 0D;
undo = undotemp;
gates = Append@gates, Pauli@@pgate, signpDDD;
gates = Append@gates, Clifford@@cgate, signcDDD<, i ++D ;

If@pgate == 4, 8zswap = -zswap, pgate = 1<D;

pgate = CallPauli@@i + 1DD;

signtemp = CallSignP@@i + 1DD;

signp = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
gates = Append@gates, Pauli@@pgate, signpDDD;
gates = Append@gates, Undo@@undo + 1, Random@Integer, 1D + 1DDD;

If@pgate == 4, 8zswap = -zswap, pgate = 1<D;

pgate = CallPauli@@i + 2DD;

signtemp = CallSignP@@i + 2DD;

signp = If@zswap ã -1, Mod@signtemp, 2D + 1, signtempD;
gates = Append@gates, Pauli@@pgate, signpDDDD;

Standard commands to define a lowering operator, indentity and dagger

 a. mathematica code: pulse generation 277
ident@dim_D := SparseArray@88Poutine$_ , Poutine$_< -> 1< , 8dim, dim<D;
destroy@dim_D :=
SparseArrayB8Poutine$_ , Maudite$_< ê; Poutine$ + 1 == Maudite$ -> Poutine$ ,

8dim, dim<F;

qbasis@dim_, n_D := Flatten@SparseArray@88n + 1, 1< -> 1< , 8dim, 1<D D;

dagger@a_D := Transpose@Conjugate@aDD;

The SU 2 operators

= dagger@s+ D;
s+ = destroy@2D;
s-
sx = s- + s+ ;
sy = Â s- - Â s+ ;
sz = - s- .s+ + s+ .s- ;

Rotations

X@q_D = MatrixExp@-Â sx q ê 2D êê Normal;

Y@q_D = MatrixExpA-Â sy q ê 2E êê Normal;
Z@q_D = MatrixExp@-Â sz q ê 2D êê Normal;
II = ident@2D êê Normal;

Numerical replacements

NRep = :X90p Ø XB F, X90m Ø XB- F, Y90p Ø YB F, Y90m Ø YB- F,

p p p p
2 2 2 2
Idp Ø II, Idm Ø II, Xp Ø X@pD, Xm Ø X@-pD, Yp Ø Y@pD, Ym Ø Y@-pD>;

NumericalCheck1@seq_D := Module@8ans, temp, test<, temp = seq ê. NRep;

For@i = 1, i § Length@tempD - 1, 8
test = temp@@1DD;

test = test.temp@@iDD <, i ++D;

ans = test.80, 1<; If@ans@@1DD ã 0, "ground", "excited"DD;
NumericalCheck2@seq_D := Module@8temp, test<, temp = seq ê. NRep;

For@i = 1, i § Length@tempD - 1, 8
test = temp@@1DD;

test = test.temp@@iDD <, i ++D;

test.80, 1<D;

ü Pulse generation sequences for single qubit randomized benchmarking

NG = 4;

lengths = 82, 3, 4, 5, 6, 8, 10, 12, 16, 20, 24, 32, 40, 48, 64, 80, 96<;
NP = 8;

NL = Length@lengthsD;
expinfo = 8<;
expsequences = 8<;
278 mathematica code: pulse generation

For@ng = 0, ng < NG, 8

exp = 0;

CallClifford = Table@Random@Integer, 1D + 1, 8i, 1, 100<D;

CallSignC = Table@Random@Integer, 1D + 1, 8i, 1, 100<D; For@np = 0, np < NP, 8
CallPauli = Table@ Random@Integer, 3D + 1, 8i, 1, 100 + 2<D;
CallSignP = Table@ Random@Integer, 1D + 1, 8i, 1, 100 + 2<D;
For@nl = 0, nl < NL, 8
numGates = lengths@@nlDD;
seq = Gates@numGatesD;

expinfo = Append@expinfo, 8exp, ng, np, nl, NumericalCheck1@seqD<D;

exp += 1;

H*Print@"Exp",exp,":"," ng=",ng," np=",np, " nl=" ,

expsequences = Append@expsequences, seqD;

<, nl ++D
nl," Result= ",NumericalCheck1@seqD,"\n"," seq=",seqD;*L

<, np ++D
<, ng ++D

1-Qubit Process Tomography

Pulse generation sequences for single qubit process tomography (pattern generation, marker generation, and
file exporting omitted)
runType = 5;

H*choose the process to be investigated *L

qubit = L;

basename = Which@ runType ã 0, "Id",

runType ã 1, "LX90p",
runType ã 2, "LX90m",
runType ã 3, "LY90p",
runType ã 4, "LY90m",
runType ã 5, "LXp",
runType ã 6, "LYp",
runType ã 7, "RX90p",
runType ã 8, "RX90m",
runType ã 9, "RY90p",
runType ã 10, "RY90m",
runType ã 11, "RXp",

D;
runType ã 12, "RYp"
 a. mathematica code: pulse generation 279
ProcTomo1QSeq@Proc_, Qubit_, m_, n_D :=

LUnitaries = 8Id, LXp, LYp, LX90p, LY90m<;

Module@8LUnitaries, RUnitaries, Prep, Meas<,

RUnitaries = 8Id, RXp, RYp, RX90p, RY90m<;

If@Qubit ã L, 8 Prep = LUnitaries@@mDD, Meas = LUnitaries@@nDD<D ;
If@Qubit ã R, 8 Prep = RUnitaries@@mDD, Meas = RUnitaries@@nDD<D ;
8Prep, Proc, Meas<D;

TomoSequences = 8<;
BuildProcTomo1Q@Proc_, Qubit_D := Module@8TomoSequences<,

For@m = 1, m § 5, m ++, 8
For@n = 1, n § 5, n ++, 8TomoSequences =
Append@TomoSequences, ProcTomo1QSeq@Proc, Qubit, m, nDD<D<D;

CalSeq = Which@ToString@qubitD ã "L", 8

TomoSequencesD;

8Id<, 8Id<, 8Id<, 8Id<, 8Id<,

8LXp<, 8LXp<, 8LXp<, 8LXp<, 8LXp<,
8LXm<, 8LXm<, 8LXm<, 8LXm<, 8LXm<,
8LYp<, 8LYp<, 8LYp<, 8LYp<, 8LYp<,
8LYm<, 8LYm<, 8LYm<, 8LYm<, 8LYm<
<,
ToString@qubitD ã "R", 8
8Id<, 8Id<, 8Id<, 8Id<, 8Id<,
8RXp<, 8RXp<, 8RXp<, 8RXp<, 8RXp<,
8RXm<, 8RXm<, 8RXm<, 8RXm<, 8RXm<,
8RYp<, 8RYp<, 8RYp<, 8RYp<, 8RYp<,
8RYm<, 8RYm<, 8RYm<, 8RYm<, 8RYm<<D;
ProcTomographySeq = BuildProcTomo1Q@
Which@ runType ã 0, Id,
runType ã 1, LX90p,
runType ã 2, LX90m,
runType ã 3, LY90p,
runType ã 4, LY90m,
runType ã 5, LXp,
runType ã 6, LYp,
runType ã 7, RX90p,
runType ã 8, RX90m,
runType ã 9, RY90p,
runType ã 10, RY90m,
runType ã 11, RXp,

D, qubitD;
runType ã 12, RYp

ProcTomographySeq = Join@CalSeq, ProcTomographySeqD;

2-Qubit State Tomography

Pulse generation sequences for two qubit state tomography (pattern generation, marker generation, and file
exporting omitted)
H*Choose the state desired *L
runType = 8;
basename = Which@ runType ã 0, "B0",
runType ã 1, "B1",
runType ã 2, "B2",
runType ã 3, "B3",
runType ã 4, "Id",
runType ã 5, "YY",
runType ã 6, "RXp",
runType ã 7, "LXp",
runType ã 8, "LXpRXp",
runType ã 9, "RY90m",
, ,
280 mathematica code: pulse generation
runType ã 9, "RY90m",
runType ã 10, "B0a",
runType ã 11, "B1a",
runType ã 12, "B2a",
runType ã 13, "B3a",
runType ã 14, "B0p0",
runType ã 15, "B0p1",
runType ã 16, "B0p2",
runType ã 17, "B0p3"D
ampFL = DACoffsetLZ; ampFR = DACoffsetRZ;
widthFL = 0; widthFR = 0;
numSigmasFL = 0.000001; numSigmasFR = 0.000001;
OneStateTomo2QSeq@Proc_, m_, opcode_D :=

Unitaries = 8Id,
Module@8Unitaries, numUnitaries, Meas<,

LXp,
RXp,
LX90p,
LX90pRX90p, LX90pRX90p, LX90pRX90p, LX90pRX90p,
LX90pRY90p, LX90pRY90p, LX90pRY90p, LX90pRY90p,
LX90pRXp,
LY90p,
LY90pRX90p, LY90pRX90p, LY90pRX90p, LY90pRX90p,
LY90pRY90p, LY90pRY90p, LY90pRY90p, LY90pRY90p,
LY90pRXp,
RX90p,
LXpRX90p,
RY90p,
LXpRY90p<;
numUnitaries = Length@UnitariesD;
Meas = Unitaries@@mDD ;
Flatten@8Proc, Meas<DD;
BuildStateTomo2QSeq@Proc_, numReps_D :=

TomoSequences = 8<;
Module@8TomoSequences, numSequences<,

For@m = 1, m § numSequences, m ++, 8

numSequences = 27;

For@n = 1, n § numReps, n ++, 8

TomoSequences =
Append@TomoSequences, OneStateTomo2QSeq@Proc, m, 1DD<D<D;

AllStateTomo2QSeq = 8
TomoSequencesD;

8Id<, 8Id<, 8Id<, 8Id<, 8Id<,

8LXp<, 8LXp<, 8LXp<, 8LXp<, 8LXp<,
8RXp<, 8RXp<, 8RXp<, 8RXp<, 8RXp<,
8LXpRXp<, 8LXpRXp<, 8LXpRXp<, 8LXpRXp<, 8LXpRXp<<;

8LY90pRY90p, LZfRZf, LY90p<,

AllStateTomo2QSeq = Join@AllStateTomo2QSeq, BuildStateTomo2QSeq@
Which@ runType § 3,
runType ã 4, Id,

H*LY90pRY90p*L, H*LY90mRY90m, LY90pRY90m,LY90mRY90p*L

runType ã 5, LY90pRY90p

runType ã 6, RXp,
runType ã 7, LXp,
runType ã 8, LXpRXp,

8LXp, RXp<,
runType ã 9, RY90m,

runType == 10, 8LY90pRY90m, LZfRZf, LY90p<,

runType ã 20,

runType == 11, 8LY90pRY90p, LZfRZf, LY90p<,

runType == 12, 8LY90mRY90m, LZfRZf, LY90p<,
runType == 13, 8LY90mRY90p, LZfRZf, LY90p<,
runType ã 14 »» runType ã 15 »» runType ã 16 »» runType ã 17,
8LZfRZf, LY90pRY90p, LZfRZf, LY90p<D, 2DD
 APPENDIX B

Mathematica code for tomography

C
ode for analyzing the raw measurements for two-qubit state tomography as well as
single-qubit process tomography. The Mathematica code is adapted from that written
and developed by postdoc Jay Gambetta. In the ‘2 qubit state tomography’ section, we
define a function Likely[] which generates a likelihood function with built in physicality
constraints and with inputs that are the raw measurements described in section 8.4. The
density matrix is then reconstructed and can be compared to theory. In ‘1 qubit process
tomography experiment’ a similar Likely[] function is used, and the χ matrix, as described
in section 6.3.5 can be obtained from a maximization of the likelihood function.

281
282 mathematica code: tomography

Initialization
Needs@"BarCharts`"D
Needs@"ErrorBarPlots`"D

ident@dim_D := SparseArray@88Poutine$_, Poutine$_< -> 1<, 8dim, dim<D;

destroy@dim_D := SparseArray@8Poutine$_, Maudite$_< ê;
Poutine$ + 1 == Maudite$ -> Sqrt@Poutine$D, 8dim, dim<D;
dagger@a_D := Transpose@Conjugate@aDD;
TensorProduct@a_, b_D := SparseArray@KroneckerProduct@a, bDD;

unflat@r_, dim_D := Transpose@Partition@r, dimD D

flat@r_D := Flatten@Transpose@rDD

Array@If@Ò > Ò2, ToExpression@"tt" <> ToString@Hn HÒ - 1L + Ò2LDD +

LowerDiagonalMatrix@f_, n_Integer? NonNegativeD :=

 ToExpression@"tt" <> ToString@Hn HÒ2 - 1L + ÒLDD, 0D &, 8n, n<D +

Array@If@Ò == Ò2, ToExpression@"tt" <> ToString@Hn HÒ - 1L + Ò2L DD, 0D &, 8n, n<D
psitorho@y_D := Outer@Times, y, Conjugate@yDD;

2 qubit state tomography

ü Operations
This section defines the Pauli algebra as well as the Likely function, with inputs from the raw measurements,
for maximization subjected to constraints.
sp = destroy@2D;
sm = dagger@spD;
sx = sp + sm;
sy = -Â sp + Â sm;
sz = - sm.sp + sp.sm;
si = ident@2D;
sii = TensorProduct@si, siD;
sxi = TensorProduct@sx, siD;
syi = TensorProduct@sy, siD;
szi = TensorProduct@sz, siD;
six = TensorProduct@si, sxD;
sxx = TensorProduct@sx, sxD;
syx = TensorProduct@sy, sxD;
szx = TensorProduct@sz, sxD;
siy = TensorProduct@si, syD;
sxy = TensorProduct@sx, syD;
syy = TensorProduct@sy, syD;
szy = TensorProduct@sz, syD;
siz = TensorProduct@si, szD;
sxz = TensorProduct@sx, szD;
syz = TensorProduct@sy, szD;
szz = TensorProduct@sz, szD;
q
U@q_, s_D := MatrixExpB-Â Normal@sDF;
2

ListPauli@rho_D := 88Tr@sii.rhoD, Tr@six.rhoD, Tr@siy.rhoD, Tr@siz.rhoD<,

8Tr@sxi.rhoD, Tr@sxx.rhoD, Tr@sxy.rhoD, Tr@sxz.rhoD<,
8Tr@syi.rhoD, Tr@syx.rhoD, Tr@syy.rhoD, Tr@syz.rhoD<,
8Tr@szi.rhoD, Tr@szx.rhoD, Tr@szy.rhoD, Tr@szz.rhoD<< ê 2;

Rho@t1_, t2_, t3_, t4_, t5_, t6_, t7_, t8_,

t9_, t10_, t11_, t12_, t13_, t14_, t15_, t16_D :=
 b. mathematica code: tomography
Rho@t1_, t2_, t3_, t4_, t5_, t6_, t7_, t8_,
283

dagger@88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0, 0<, 8t11 + Â t12, t7 + Â t8, t3, 0<,
t9_, t10_, t11_, t12_, t13_, t14_, t15_, t16_D := Simplify@

8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<<D.88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0,
0<, 8t11 + Â t12, t7 + Â t8, t3, 0<, 8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<< ê
Tr@dagger@88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0, 0<, 8t11 + Â t12, t7 + Â t8, t3, 0<,
8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<<D.88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0,
0<, 8t11 + Â t12, t7 + Â t8, t3, 0<, 8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<<D,
Assumptions Ø 8 8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11,
t12, t13, t14, t15, t16< œ Reals<D;

Likely@t1_, t2_, t3_, t4_, t5_, t6_, t7_, t8_, t9_, t10_, t11_,
t12_, t13_, t14_, t15_, t16_, r1_, r2_, r3_, r4_, r5_, r6_, r7_, r8_,
r9_, r10_, r11_, r12_, r13_, r14_, r15_, r16_, b1_, b2_, b12_D :=
H-1 + r16L2 + I-r12 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 +
t32 + t42 + t52 + t62 + t72 + t82 + t92 M + t12 b1 - t102 b1 + t112 b1 +
t122 b1 + t132 b1 + t142 b1 + t152 b1 + t162 b1 + t22 b1 - t32 b1 - t42 b1 +
t52 b1 + t62 b1 + t72 b1 + t82 b1 - t92 b1 - 2 t14 t15 b12 + 2 t13 t16 b12 -
2 t10 t4 b12 + 2 t2 t6 b12 + 2 t12 t7 b12 - 2 t11 t8 b12 - 2 t14 t15 b2 +
2 t13 t16 b2 + 2 t10 t4 b2 + 2 t2 t6 b2 + 2 t12 t7 b2 - 2 t11 t8 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

Ir13 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + t12 b1 - t102 b1 + t112 b1 + t122 b1 +
t132 b1 + t142 b1 + t152 b1 + t162 b1 + t22 b1 - t32 b1 - t42 b1 + t52 b1 +
t62 b1 + t72 b1 + t82 b1 - t92 b1 - 2 t14 t15 b12 + 2 t13 t16 b12 -
2 t10 t4 b12 + 2 t2 t6 b12 + 2 t12 t7 b12 - 2 t11 t8 b12 + 2 t14 t15 b2 -
2 t13 t16 b2 - 2 t10 t4 b2 - 2 t2 t6 b2 - 2 t12 t7 b2 + 2 t11 t8 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

I-r14 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + t12 b1 - t102 b1 + t112 b1 + t122 b1 +
t132 b1 + t142 b1 + t152 b1 + t162 b1 + t22 b1 - t32 b1 - t42 b1 + t52 b1 +
t62 b1 + t72 b1 + t82 b1 - t92 b1 - 2 t13 t15 b12 - 2 t14 t16 b12 -
2 t2 t5 b12 - 2 t11 t7 b12 - 2 t12 t8 b12 + 2 t4 t9 b12 - 2 t13 t15 b2 -
2 t14 t16 b2 - 2 t2 t5 b2 - 2 t11 t7 b2 - 2 t12 t8 b2 - 2 t4 t9 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

Ir15 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + t12 b1 - t102 b1 + t112 b1 + t122 b1 +
t132 b1 + t142 b1 + t152 b1 + t162 b1 + t22 b1 - t32 b1 - t42 b1 + t52 b1 +
t62 b1 + t72 b1 + t82 b1 - t92 b1 - 2 t13 t15 b12 - 2 t14 t16 b12 -
2 t2 t5 b12 - 2 t11 t7 b12 - 2 t12 t8 b12 + 2 t4 t9 b12 + 2 t13 t15 b2 +
2 t14 t16 b2 + 2 t2 t5 b2 + 2 t11 t7 b2 + 2 t12 t8 b2 + 2 t4 t9 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

Ir11 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + 2 t11 t3 b1 + 2 t13 t4 b1 + 2 t15 t9 b1 +
- + - + +
284 mathematica code: tomography
2 t10 t16 Hb1 - b12L - 2 t11 t3 b12 + 2 t13 t4 b12 - 2 t15 t9 b12 + t12 b2 +
t102 b2 + t112 b2 + t122 b2 - t132 b2 - t142 b2 + t152 b2 + t162 b2 -
t22 b2 + t32 b2 - t42 b2 + t52 b2 + t62 b2 - t72 b2 - t82 b2 + t92 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

Ir7 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 +
t62 + t72 + t82 + t92 M - 2 t12 t3 b1 - 2 t14 t4 b1 - 2 t16 t9 b1 +
2 t10 t15 Hb1 - b12L + 2 t12 t3 b12 - 2 t14 t4 b12 + 2 t16 t9 b12 + t12 b2 +
t102 b2 + t112 b2 + t122 b2 - t132 b2 - t142 b2 + t152 b2 + t162 b2 -
t22 b2 + t32 b2 - t42 b2 + t52 b2 + t62 b2 - t72 b2 - t82 b2 + t92 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

I-r4 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + 2 t12 t3 b1 + 2 t14 t4 b1 + 2 t16 t9 b1 +
2 t12 t3 b12 - 2 t14 t4 b12 + 2 t16 t9 b12 - 2 t10 t15 Hb1 + b12L + t12 b2 +
t102 b2 + t112 b2 + t122 b2 - t132 b2 - t142 b2 + t152 b2 + t162 b2 -
t22 b2 + t32 b2 - t42 b2 + t52 b2 + t62 b2 - t72 b2 - t82 b2 + t92 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

I-r8 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M - 2 t11 t3 b1 - 2 t13 t4 b1 - 2 t15 t9 b1 -
2 t11 t3 b12 + 2 t13 t4 b12 - 2 t15 t9 b12 - 2 t10 t16 Hb1 + b12L + t12 b2 +
t102 b2 + t112 b2 + t122 b2 - t132 b2 - t142 b2 + t152 b2 + t162 b2 -
t22 b2 + t32 b2 - t42 b2 + t52 b2 + t62 b2 - t72 b2 - t82 b2 + t92 b2M í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 +
t42 + t52 + t62 + t72 + t82 + t92 M +
2

Ir1 - IIt132 + t142 + t22 + t72 + t82 M Hb1 - b12 - b2LM ë It12 + t102 + t112 + t122 +
t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
IIt102 + t32 + t92 M Hb1 + b12 - b2LM ë It12 + t102 + t112 + t122 + t132 +
t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
It42 Hb1 - b12 + b2LM ë It12 + t102 + t112 + t122 + t132 + t142 + t152 +
t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M -
IIt12 + t112 + t122 + t152 + t162 + t52 + t62 M Hb1 + b12 + b2LM ë It12 + t102 + t112 +
t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 MM +
2

Ir3 - IIt12 + t112 + t122 + t152 + t162 + t52 + t62 M Hb1 - b12 - b2LM ë It12 + t102 + t112 +
t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
It42 Hb1 + b12 - b2LM ë It12 + t102 + t112 + t122 + t132 + t142 + t152 +
t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
IIt102 + t32 + t92 M Hb1 - b12 + b2LM ë It12 + t102 + t112 + t122 + t132 +
t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M -
IIt132 + t142 + t22 + t72 + t82 M Hb1 + b12 + b2LM ë It12 + t102 + t112 + t122 +
t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 MM +
2
 b. mathematica code: tomography 285
Ir2 - It42 Hb1 - b12 - b2LM ë It12 + t102 + t112 + t122 + t132 + t142 + t152 +
t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
IIt12 + t112 + t122 + t152 + t162 + t52 + t62 M Hb1 + b12 - b2LM ë It12 + t102 + t112 +
t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M +
IIt132 + t142 + t22 + t72 + t82 M Hb1 - b12 + b2LM ë It12 + t102 + t112 + t122 +
t132 + t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M -
IIt102 + t32 + t92 M Hb1 + b12 + b2LM ë It12 + t102 + t112 + t122 + t132 +
t142 + t152 + t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 MM +
2

Ir10 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + 2 Ht10 t16 b1 + t11 t3 b1 + t13 t4 b1 +
t15 t9 b1 - t10 t14 b12 - t15 t4 b12 - t3 t7 b12 - t13 t9 b12 +
t13 t15 b2 + t14 t16 b2 + t2 t5 b2 + t11 t7 b2 + t12 t8 b2 + t4 t9 b2LM í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M +
2

Ir6 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + 2 Ht10 t15 b1 - t12 t3 b1 - t14 t4 b1 -
t16 t9 b1 - t10 t13 b12 + t16 t4 b12 + t3 t8 b12 + t14 t9 b12 +
t13 t15 b2 + t14 t16 b2 + t2 t5 b2 + t11 t7 b2 + t12 t8 b2 + t4 t9 b2LM í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M +
2

Ir9 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M + 2 Ht11 t3 b1 + t13 t4 b1 + t15 t9 b1 +
t16 t4 b12 - t3 t8 b12 - t14 t9 b12 + t14 t15 b2 - t13 t16 b2 -
t2 t6 b2 - t12 t7 b2 + t11 t8 b2 + t10 Ht16 b1 + t13 b12 - t4 b2LLM í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M +
2

Ir5 It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M - 2 Ht12 t3 b1 + t14 t4 b1 + t16 t9 b1 -
t15 t4 b12 + t3 t7 b12 + t13 t9 b12 - t14 t15 b2 + t13 t16 b2 +
t2 t6 b2 + t12 t7 b2 - t11 t8 b2 + t10 H-t15 b1 + t14 b12 + t4 b2LLM í
2

It12 + t102 + t112 + t122 + t132 + t142 + t152 + t162 + t22 + t32 + t42 +
t52 + t62 + t72 + t82 + t92 M
2

Define the theoretical measurements, given experimental amplitudes that describe the joint measurement

Clear@b1, b2 , b12 D;
Clear@a1, a2 , a12 D;
M = Hb1 szi + b2 siz + b12 szzL;

m2 = dagger@U@p , sxiDD.M .U@p , sxi D;

m1 = M;

m3 = dagger@U@p , sixDD.M .U@p , six D;

m4 = daggerBUB , sxiFF.M .UB , sxi F;
p p
2 2
m5 = daggerBUB , sxi + sixFF.M .UB , sxi + six F;
p p
2 2
286 mathematica code: tomography
, sxi + siy F;
p p
m6 = daggerBUB , sxi + siyFF.M .UB
2 2
p p
m7 = daggerBUB , sxi + 2 sixFF.M .UB , sxi + 2 sixF;
2 2
p p
m8 = daggerBUB , syiFF.M .UB , syiF;
2 2
, syi + six F;
p p
m9 = daggerBUB , syi + sixFF.M .UB
2 2
syi + siyFF.M .UB , syi + siy F;
p p
m10 = daggerBUB ,
2 2
syi + 2 sixFF.M .UB , syi + 2 six F;
p p
m11 = daggerBUB ,
2 2
sixFF.M .UB , six F;
p p
m12 = daggerBUB ,
2 2
six + 2 sxiFF.M .UB , six + 2 sxi F;
p p
m13 = daggerBUB ,
2 2
siyFF.M .UB , siy F;
p p
m14 = daggerBUB ,
2 2
p p
m15 = daggerBUB , siy + 2 sxiFF.M .UB , siy + 2 sxiF;
2 2
m16 = sii;

P = Ha1 szi + a2 siz + a12 szzL;

p2 = dagger@U@p , sxiDD.P.U@p , sxi D;

p1 = P;

p3 = dagger@U@p , sixDD.P .U@p , six D;

p4 = daggerBUB , sxiFF.P .UB , sxi F;
p p
2 2
p5 = daggerBUB , sxi + sixFF.P .UB , sxi + six F;
p p
2 2
p6 = daggerBUB , sxi + siyFF.P.UB , sxi + siy F;
p p
2 2
p p
p7 = daggerBUB , sxi + 2 sixFF.P .UB , sxi + 2 sixF;
2 2
p p
p8 = daggerBUB , syiFF.P.UB , syiF;
2 2
p9 = daggerBUB , syi + sixFF.P .UB , syi + six F;
p p
2 2
p10 = daggerBUB , syi + siyFF.P .UB , syi + siy F;
p p
2 2
p11 = daggerBUB , syi + 2 sixFF.P .UB , syi + 2 six F;
p p
2 2
p12 = daggerBUB , sixFF.P.UB , six F;
p p
2 2
p13 = daggerBUB , six + 2 sxiFF.P .UB , six + 2 sxi F;
p p
2 2
p14 = daggerBUB , siyFF.P .UB , siy F;
p p
2 2
p p
p15 = daggerBUB , siy + 2 sxiFF.P .UB , siy + 2 sxiF;
2 2
p16 = sii;
 b. mathematica code: tomography 287
8Tr@m1.siiD, Tr@m1.sxiD, Tr@m1.syiD, Tr@m1.sziD, Tr@m1.sixD,
1
RowAforI@m1_D :=
4
Tr@m1.sxxD, Tr@m1.syxD, Tr@m1.szxD, Tr@m1.siyD, Tr@m1.sxyD, Tr@m1.syyD,
Tr@m1.szyD, Tr@m1.sizD, Tr@m1.sxzD, Tr@m1.syzD, Tr@m1.szzD<;

8i, 1, 16<D;
AforI = Table@FullSimplify@RowAforI@ToExpression@"m" <> ToString@ iDDDD,

AiforI = Inverse@AforID;
RowAforQ@p1_D :=

8Tr@p1.siiD, Tr@p1.sxiD, Tr@p1.syiD, Tr@p1.sziD, Tr@p1.sixD,

1
4
Tr@p1.sxxD, Tr@p1.syxD, Tr@p1.szxD, Tr@p1.siyD, Tr@p1.sxyD, Tr@p1.syyD,
Tr@p1.szyD, Tr@p1.sizD, Tr@p1.sxzD, Tr@p1.syzD, Tr@p1.szzD<;

8i, 1, 16<D;
AforQ = Table@FullSimplify@RowAforQ@ToExpression@"p" <> ToString@ iDDDD,

AiforQ = Inverse@AforQD;
MatrixRank@8Flatten@m1D, Flatten@m2D, Flatten@m3D,
Flatten@m4D, Flatten@m5D, Flatten@m6D, Flatten@m7D, Flatten@m8D,
Flatten@m9D, Flatten@m10D, Flatten@m11D, Flatten@m12D,
Flatten@m13D, Flatten@m14D, Flatten@m15D, Flatten@m16D<D
MatrixRank@8Flatten@m1D, Flatten@m2D, Flatten@m3D, Flatten@m4D,
Flatten@m5D, Flatten@m6D, Flatten@m7D, Flatten@m8D, Flatten@m9D,

Flatten@m14D, Flatten@m15D, Flatten@m16D< ê. b12 Ø 0D

Flatten@m10D, Flatten@m11D, Flatten@m12D, Flatten@m13D,

Define some metrics, including concurrence, entanglement of formation, and fidelity to the targeted state.

,
Con@r_D := ModuleA8l<, l =
ChopASortA Eigenvalues@Hr.syy.Conjugate@rD.syy LD, Re@Ò1D > Re@Ò2D &E, 10-7 E;
Max@0, l@@1DD - l@@2DD - l@@3DD - l@@4DDDE;

F-
1+ 1 - c2 1+ 1 - c2
Ef@c_D := - LogB2,
2 2

F;
1+ 1 - c2 1+ 1 - c2
1- LogB2, 1 -
2 2

Fidelity@r1_, r2_D := ChopB

F.r2.MatrixPowerBr1, F, FF F, 10-8 F;
2
1 1 1
ReBTrBMatrixPowerBMatrixPowerBr1,
2 2 2

ü 2 Qubit state tomography from experimental data - 2 Input sets

After inputting the raw measurements in both the I and Q quadratures, we reconstruct the physical density
matrix with a constrained maximization of the Likely function.
datapath = "C:\\Experiments\\LatestTomography\\";
thisNum = 0;

suffix = Which@thisNum ã 0, "", thisNum > 0, "_" <> ToString@thisNumDD;

ST2QvecI1 = Import@datapath <> "ST2QvecI" <> suffix <> ".txt", "Table"D;

ST2QvecQ1 = Import@datapath <> "ST2QvecQ" <> suffix <> ".txt", "Table"D;
ST2QvecI = Flatten@ST2QvecQ1D;
ST2QvecQ = Flatten@ST2QvecQ1D;
288 mathematica code: tomography
ST2QvecQ = Flatten@ST2QvecQ1D;

b1 = ST2QvecI@@1DD; b2 = ST2QvecI@@2DD; b12 = ST2QvecI@@3DD;

a1 = ST2QvecQ@@1DD; a2 = ST2QvecQ@@2DD; a12 = ST2QvecQ@@3DD;
m1i = ST2QvecI@@4DD; m2i = ST2QvecI@@5DD;
m3i = ST2QvecI@@6DD; m4i = ST2QvecI@@7DD;
m5i = ST2QvecI@@8DD; m6i = ST2QvecI@@9DD;
m7i = ST2QvecI@@10DD; m8i = ST2QvecI@@11DD;
m9i = ST2QvecI@@12DD; m10i = ST2QvecI@@13DD;
m11i = ST2QvecI@@14DD; m12i = ST2QvecI@@15DD;
m13i = ST2QvecI@@16DD; m14i = ST2QvecI@@17DD;
m15i = ST2QvecI@@18DD; m16i = ST2QvecI@@19DD;
m1q = ST2QvecQ@@4DD; m2q = ST2QvecQ@@5DD;
m3q = ST2QvecQ@@6DD; m4q = ST2QvecQ@@7DD;
m5q = ST2QvecQ@@8DD; m6q = ST2QvecQ@@9DD;
m7q = ST2QvecQ@@10DD; m8q = ST2QvecQ@@11DD;
m9q = ST2QvecQ@@12DD; m10q = ST2QvecQ@@13DD;
m11q = ST2QvecQ@@14DD; m12q = ST2QvecQ@@15DD;
m13q = ST2QvecQ@@16DD; m14q = ST2QvecQ@@17DD;
m15q = ST2QvecQ@@18DD; m16q = ST2QvecQ@@19DD;

q2 = 1; j = 1;
epsL = -0.0;
epsR = -0.00;
yTheo = Bell2@0, 0, 0, 0D;

M = Hb1 szi + b2 siz + b12 szzL;

RhoTheo = psitorho@yTheoD;

n2 = dagger@U@p , H1 + epsLL sxiDD.M .U@p , H1 + epsLL sxi D;

n1 = M;

n3 = dagger@U@p , q2 H1 + epsRL sixDD.M .U@p , q2 H1 + epsRL six D;

n4 = daggerBUB , H1 + epsLL sxiFF.M .UB , H1 + epsLL sxi F;
p p
2 2
n5 = daggerBUB , H1 + epsLL sxi + q2 H1 + epsRL sixFF.
p
2
M .UB , H1 + epsLL sxi + q2 H1 + epsRL six F;
p
2
n6 = daggerBUB , H1 + epsLL sxi + j q2 H1 + epsRL siyFF.M .
p
2
UB , H1 + epsLL sxi + j q2 H1 + epsRL siy F;
p
2
n7 = daggerBUB , H1 + epsLL sxi + 2 q2 H1 + epsRL sixFF.M .
p
2
UB , H1 + epsLL sxi + 2 q2 H1 + epsRL sixF;
p
2
n8 = daggerBUB , j H1 + epsLL syiFF.M .UB , j H1 + epsLL syiF;
p p
2 2
n9 = daggerBUB , j H1 + epsLL syi + q2 H1 + epsRL sixFF.
p
2
M .UB , j H1 + epsLL syi + q2 H1 + epsRL six F;
p
2
n10 = daggerBUB , j H1 + epsLL syi + j q2 H1 + epsRL siyFF.
p
2
M .UB , j H1 + epsLL syi + j q2 H1 + epsRL siy F;
p
2
n11 = daggerBUB , j H1 + epsLL syi + 2 q2 H1 + epsRL sixFF.
p
2
M .UB , j H1 + epsLL syi + 2 q2 H1 + epsRL six F;
p
2
 b. mathematica code: tomography 289
, q2 H1 + epsRL sixFF.M .UB , q2 H1 + epsRL six F;
p p
n12 = daggerBUB
2 2
, q2 H1 + epsRL six + 2 H1 + epsLL sxiFF.
p
n13 = daggerBUB
2
, q2 H1 + epsRL six + H1 + epsLL 2 sxi F;
p
M .UB
2
, j q2 H1 + epsRL siyFF.M .UB , j q2 H1 + epsRL siy F;
p p
n14 = daggerBUB
2 2
, j q2 H1 + epsRL siy + 2 H1 + epsLL sxiFF.
p
n15 = daggerBUB
2
, j q2 H1 + epsRL siy + H1 + epsLL 2 sxiF;
p
M .UB
2
n16 = sii;

P = Ha1 szi + a2 siz + a12 szzL;

p2 = dagger@U@p , H1 + epsLL sxiDD.P.U@p , H1 + epsLL sxi D;

p1 = P;

p3 = dagger@U@p , q2 H1 + epsRL sixDD.P .U@p , q2 H1 + epsRL six D;

p4 = daggerBUB , H1 + epsLL sxiFF.P .UB , H1 + epsLL sxi F;
p p
2 2
p5 = daggerBUB , H1 + epsLL sxi + q2 H1 + epsRL sixFF.
p
2
P .UB , H1 + epsLL sxi + q2 H1 + epsRL six F;
p
2
p6 = daggerBUB , H1 + epsLL sxi + j q2 H1 + epsRL siyFF.P .
p
2
UB , H1 + epsLL sxi + j q2 H1 + epsRL siy F;
p
2
p7 = daggerBUB , H1 + epsLL sxi + 2 q2 H1 + epsRL sixFF.P .
p
2
UB , H1 + epsLL sxi + 2 q2 H1 + epsRL sixF;
p
2
p8 = daggerBUB , j H1 + epsLL syiFF.P.UB , j H1 + epsLL syiF;
p p
2 2
p9 = daggerBUB , j H1 + epsLL syi + q2 H1 + epsRL sixFF.
p
2
P.UB , j H1 + epsLL syi + q2 H1 + epsRL six F;
p
2
p10 = daggerBUB , j H1 + epsLL syi + j q2 H1 + epsRL siyFF.
p
2
P.UB , j H1 + epsLL syi + j q2 H1 + epsRL siy F;
p
2
p11 = daggerBUB , j H1 + epsLL syi + 2 q2 H1 + epsRL sixFF.
p
2
P.UB , j H1 + epsLL syi + 2 q2 H1 + epsRL six F;
p
2
p12 = daggerBUB , q2 H1 + epsRL sixFF.P.UB , q2 H1 + epsRL six F;
p p
2 2
p13 = daggerBUB , q2 H1 + epsRL six + 2 H1 + epsLL sxiFF.
p
2
P.UB , q2 H1 + epsRL six + H1 + epsLL 2 sxi F;
p
2
290 mathematica code: tomography
, j q2 H1 + epsRL siyFF.P.UB , j q2 H1 + epsRL siy F;
p p
p14 = daggerBUB
2 2
, j q2 H1 + epsRL siy + 2 H1 + epsLL sxiFF.
p
p15 = daggerBUB
2
, j q2 H1 + epsRL siy + H1 + epsLL 2 sxiF;
p
P.UB
2
p16 = sii;

H* Simulated measurement results *L

t1i1 = Re@Tr@n1.RhoTheoDD; t2i1 = Re@Tr@n2.RhoTheoDD;
t3i1 = Re@Tr@n3.RhoTheoDD; t4i1 = Re@Tr@n4.RhoTheoDD;
t5i1 = Re@Tr@n5.RhoTheoDD; t6i1 = Re@Tr@n6.RhoTheoDD;
t7i1 = Re@Tr@n7.RhoTheoDD; t8i1 = Re@Tr@n8.RhoTheoDD;
t9i1 = Re@Tr@n9.RhoTheoDD; t10i1 = Re@Tr@n10.RhoTheoDD;
t11i1 = Re@Tr@n11.RhoTheoDD; t12i1 = Re@Tr@n12.RhoTheoDD;

t15i1 = Re@Tr@n15.RhoTheoDD; t16i1 = Re@Tr@n16.RhoTheoD D;

t13i1 = Re@Tr@n13.RhoTheoDD; t14i1 = Re@Tr@n14.RhoTheoDD;

t1q1 = Re@Tr@p1.RhoTheoDD; t2q1 = Re@Tr@p2.RhoTheoDD;

t3q1 = Re@Tr@p3.RhoTheoDD; t4q1 = Re@Tr@p4.RhoTheoDD;
t5q1 = Re@Tr@p5.RhoTheoDD; t6q1 = Re@Tr@p6.RhoTheoDD;
t7q1 = Re@Tr@p7.RhoTheoDD; t8q1 = Re@Tr@p8.RhoTheoDD;
t9q1 = Re@Tr@p9.RhoTheoDD; t10q1 = Re@Tr@p10.RhoTheoDD;
t11q1 = Re@Tr@p11.RhoTheoDD; t12q1 = Re@Tr@p12.RhoTheoDD;

t15q1 = Re@Tr@p15.RhoTheoDD; t16q1 = Re@Tr@p16.RhoTheoD D;

t13q1 = Re@Tr@p13.RhoTheoDD; t14q1 = Re@Tr@p14.RhoTheoDD;

ExpVecI = 8m1i, m2i, m3i, m4i, m5i, m6i,

ExpVecQ = 8m1q, m2q, m3q, m4q, m5q, m6q, m7q, m8q, m9q,
m7i, m8i, m9i, m10i, m11i, m12i, m13i, m14i, m15i, m16i<;

TheoryVecI = 8t1i1, t2i1, t3i1, t4i1, t5i1, t6i1, t7i1, t8i1,

m10q, m11q, m12q, m13q, m14q, m15q, m16q<;

TheoryVecQ = 8t1q1, t2q1, t3q1, t4q1, t5q1, t6q1, t7q1, t8q1,

t9i1, t10i1, t11i1, t12i1, t13i1, t14i1, t15i1, t16i1<;

t9q1, t10q1, t11q1, t12q1, t13q1, t14q1, t15q1, t16q1<;

solExp = NMinimize@
Chop@
Likely@t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16,
m1i, m2i, m3i, m4i, m5i, m6i, m7i, m8i,
m9i, m10i, m11i, m12i, m13i, m14i, m15i, m16i, b1, b2, b12D +
Likely@t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16,
m1q, m2q, m3q, m4q, m5q, m6q, m7q, m8q,

8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16<D;
m9q, m10q, m11q, m12q, m13q, m14q, m15q, m16q, a1, a2, a12DD,

t8, t9, t10, t11, t12, t13, t14, t15, t16D ê. solExp@@2DD;
RhoExp = Rho@t1, t2, t3, t4, t5, t6, t7,

solTheo = NMinimize@
Chop@
Likely@t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16,
t1i1, t2i1, t3i1, t4i1, t5i1, t6i1, t7i1, t8i1, t9i1,
t10i1, t11i1, t12i1, t13i1, t14i1, t15i1, t16i1, b1, b2, b12D +
Likely@t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16,
t1q1, t2q1, t3q1, t4q1, t5q1, t6q1, t7q1, t8q1, t9q1,

8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16<D;
t10q1, t11q1, t12q1, t13q1, t14q1, t15q1, t16q1, a1, a2, a12DD,

t8, t9, t10, t11, t12, t13, t14, t15, t16D ê. solTheo@@2DD;
RhoTheo2 = Rho@t1, t2, t3, t4, t5, t6, t7,
 t8, t9, t10, t11, t12, t13, t14, t15, t16D ê. solTheo@@2DD;
b. mathematica code: tomography 291

H* reconstructing the state *L

8Chop@Tr@RhoExpDD, Chop@Tr@RhoExp.RhoExpDD<
8Con@RhoExpD, Ef@Con@RhoExpDD<

H* Fidelity of state to »10> and to »01> *L

Fidelity@RhoExp, RhoTheo2D

y = 81, 0, 0, 0<; RhoT1 = psitorho@yD;

y = 80, 1, 0, 0<; RhoT2 = psitorho@yD;
y = 80, 0, 1, 0<; RhoT3 = psitorho@yD;
y = 80, 0, 0, 1<; RhoT4 = psitorho@yD;
8Fidelity@RhoExp, RhoT1D, Fidelity@RhoExp, RhoT2D,
Fidelity@RhoExp, RhoT3D, Fidelity@RhoExp, RhoT4D <
88Chop@Tr@RhoExpDD<, 8Chop@Tr@RhoExp.RhoExpDD<,
8Con@RhoExpD<, 8Ef@Con@RhoExpDD<, 8Fidelity@RhoExp, RhoTheoD<<;

H* Simulated measurement results *L

t1i2 = Re@Tr@n1.RhoExpDD;
t2i2 = Re@Tr@n2.RhoExpDD;
t3i2 = Re@Tr@n3.RhoExpDD;
t4i2 = Re@Tr@n4.RhoExpDD;
t5i2 = Re@Tr@n5.RhoExpDD;
t6i2 = Re@Tr@n6.RhoExpDD;
t7i2 = Re@Tr@n7.RhoExpDD;
t8i2 = Re@Tr@n8.RhoExpDD;
t9i2 = Re@Tr@n9.RhoExpDD;
t10i2 = Re@Tr@n10.RhoExpDD;
t11i2 = Re@Tr@n11.RhoExpDD;
t12i2 = Re@Tr@n12.RhoExpDD;
t13i2 = Re@Tr@n13.RhoExpDD;
t14i2 = Re@Tr@n14.RhoExpDD;
t15i2 = Re@Tr@n15.RhoExpDD;
t16i2 = Re@Tr@n16.RhoExpDD;

H* Simulated measurement results *L

t1q2 = Re@Tr@p1.RhoExpD D;
t2q2 = Re@Tr@p2.RhoExpDD;
t3q2 = Re@Tr@p3.RhoExpDD;
t4q2 = Re@Tr@p4.RhoExpDD;
t5q2 = Re@Tr@p5.RhoExpDD;
t6q2 = Re@Tr@p6.RhoExpDD;
t7q2 = Re@Tr@p7.RhoExpDD;
t8q2 = Re@Tr@p8.RhoExpDD;
t9q2 = Re@Tr@p9.RhoExpDD;
t10q2 = Re@Tr@p10.RhoExpDD;
t11q2 = Re@Tr@p11.RhoExpDD;
t12q2 = Re@Tr@p12.RhoExpDD;
t13q2 = Re@Tr@p13.RhoExpDD;
t14q2 = Re@Tr@p14.RhoExpDD;
t15q2 = Re@Tr@p15.RhoExpDD;
t16q2 = Re@Tr@p16.RhoExpDD;

TheoryVecI2 = 8t1i2, t2i2, t3i2, t4i2, t5i2, t6i2, t7i2,

TheoryVecQ2 = 8t1q2, t2q2, t3q2, t4q2, t5q2, t6q2, t7q2, t8q2,

t8i2, t9i2, t10i2, t11i2, t12i2, t13i2, t14i2, t15i2, t16i2<;

t9q2, t10q2, t11q2, t12q2, t13q2, t14q2, t15q2, t16q2<;

PlotMax = Max@Abs@b1D + Abs@b2D + Abs@b12D, Abs@a1D + Abs@a2D + Abs@a12DD;

PlotMin = -PlotMax;

8RGBColor@1, 0, 0D, RGBColor@0, 1, 0D, RGBColor@1, 0, 0D, RGBColor@0, 0, 1D<,

ListPlot@8ExpVecI, TheoryVecI, ExpVecQ, TheoryVecQ<, PlotStyle Ø

PlotJoined Ø True, PlotRange Ø 8PlotMin, PlotMax<, GridLines Ø Automatic,

, D
 PlotJoined Ø True, PlotRange Ø 8PlotMin, PlotMax<,
292 mathematica code: tomography
Frame Ø True, FrameLabel Ø 8"Measurement number", "Amplitude"<D
GridLines Ø Automatic,

BarChart3D@Re@RhoExpD, PlotRange -> 8All, All, 8-1, 1<<,

GraphicsRow@8

BarStyle Ø Directive@Glow@BlueD, Opacity@0.9DD, BarEdgeStyle Ø Blue,

BarChart3D@Im@RhoExpD, PlotRange -> 8All, All, 8-1, 1<<,

BarSpacing Ø .15, BarEdges Ø True, Boxed Ø TrueD,

BarStyle Ø Directive@Glow@BlueD, Opacity@0.9DD, BarEdgeStyle Ø Blue,

<D;
BarSpacing Ø .15, BarEdges Ø True, Boxed Ø TrueD

BarChart3D@Re@RhoExpD, PlotRange -> 8All, All, 8-1, 1<< D,

GraphicsRow@8

BarChart3D@Im@RhoExpD, PlotRange -> 8All, All, 8-1, 1<<D

<D

BarChart3D@Re@RhoTheo2D, PlotRange -> 8All, All, 8-1, 1<< D,

GraphicsRow@8

BarChart3D@Im@RhoTheo2D, PlotRange -> 8All, All, 8-1, 1<<D

<D

BarChart3D@Re@RhoExpD, PlotRange -> 8All, All, 8-1, 1<<,

GraphicsRow@8

BarStyle Ø Directive@Glow@BlueD, Opacity@0.7DD, BarEdgeStyle Ø Blue,

Ticks Ø 881, 2, 3, 4<, 81, 2, 3, 4<, 8-1, -0.5, 0, 0.5, 1<<,

BarSpacing Ø .15, BarEdges Ø True, Boxed Ø True,

ViewPoint Ø 81.3, -2.4, 1.5<,

FaceGrids Ø 8 8 80, -1, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < <,
8 80, 1, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < < ,
8 81, 0, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < < ,
8 8-1, 0, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < <<,

BarChart3D@Im@RhoExpD, PlotRange -> 8All, All, 8-1, 1<<,

FaceGridsStyle Ø Directive@Thin, GrayDD,

BarStyle Ø Directive@Glow@BlueD, Opacity@0.7DD, BarEdgeStyle Ø Blue,

Ticks Ø 881, 2, 3, 4<, 81, 2, 3, 4<, 8-1, -0.5, 0, 0.5, 1<<,

BarSpacing Ø .15, BarEdges Ø True, Boxed Ø True,

ViewPoint Ø 81.3, -2.4, 1.5<,

FaceGrids Ø 8 8 80, -1, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < <,
8 80, 1, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < < ,
8 81, 0, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < < ,
8 8-1, 0, 0<, 88<, 88-0.5, Dashed<, 80.5, Dashed<< < < <,

<D;
FaceGridsStyle Ø Directive@Thin, GrayDD

1 qubit process tomography experiment

This section shows the maximum-likelihood estimation of the c matrix for a single-qubit quantum process
tomography experiment.

ü Operations for QPT

dagger@88t1, 0<, 8t3 + Â t4, t2<<D.88t1, 0<, 8t3 + Â t4, t2<<

Rho@t1_, t2_, t3_, t4_D :=

Tr@dagger@88t1, 0<, 8t3 + Â t4, t2<<D.88t1, 0<, 8t3 + Â t4, t2<<D

SimplifyB ,

Assumptions Ø 8 8t1, t2, t3, t4< œ Reals<F

 b. mathematica code: tomography 293
sp = destroy@2D;
sm = dagger@spD;
sx = sp + sm;
sy = -Â sp + Â sm;
sz = - sm.sp + sp.sm;
si = ident@2D;

q
U@q_, s_D := MatrixExpB-Â Normal@sDF;
2

H* the basis of physical measurements states *L

rj1 = 880, 0<, 80, 1<<;
rj2 = U@p, sxD.rj1.dagger@U@p, sxDD
p p
rj3 = UB , sxF.rj1.daggerBUB , sxFF
2 2

p p
rj4 = UB , syF.rj1.daggerBUB , syFF
2 2

-p -p
rj5 = UB , sxF.rj1.daggerBUB , sxFF
2 2

-p p
rj6 = UB , syF.rj1.daggerBUB- , syFF
2 2
Likely@t1_, t2_, t3_, t4_, pg_, pe_, py_, px_, V_D :=

+ H-2 + peL pe + pg2 + H-1 + pxL px + H-1 + pyL py +

1 3
2V 2

-t12 t22 + t24 t2 HH-1 + 2 pe - 2 pgL t2 + H-1 + 2 pxL t3 + t4 - 2 py t4L

+
It12 + t22 + t32 + t4 M 2 2 t12 + t22 + t32 + t42

sol = NMinimize@Likely@t1, t2, t3, t4, xm, ym, zm, im, VD, 8t1, t2, t3, t4<D;
MLE@xm_, ym_, zm_, im_, V_D := Module@8t1, t2, t3, t4, sol<,

flat@Rho@t1, t2, t3, t4D ê. sol@@2DDDD

LMat @rjm1_, rjm2_, rjm3_, rjm4_D :=

ModuleB8rjmtemp, rktemp<, rjmtemp = 8rjm1, rjm2, rjm3, rjm4<;

rktemp = : >;
flat@siD flat@sxD flat@syD flat@szD
, , ,
2 2 2 2
Table@Conjugate@rktemp@@kDDD.rjmtemp@@jDD, 8k, 1, 4<, 8j, 1, 4<DF

BMat@rj1_, rj2_, rj3_, rj4_D :=

ModuleB8rjtemp, rktemp, S<, rjtemp = 8rj1, rj2, rj3, rj4<;

rktemp = : >; S = 8si, sx, sy, sz<;

flat@siD flat@sxD flat@syD flat@szD
, , ,
2 2 2 2

flat@S@@mDD.rjtemp@@Floor@Hp - 1L ê 4 + 1DDD.S@@nDDD,
Table@Flatten@Table@Conjugate@rktemp@@Mod@p - 1, 4D + 1DDD.

8m, 1, 4<, 8n, 1, 4<DD, 8p, 1, 16<DF;

294 mathematica code: tomography
ChiMat@rjm1_, rjm2_, rjm3_, rjm4_, rj1_, rj2_, rj3_, rj4_D :=
ModuleB8rjtemp, rjmtemp, rktemp, S, l, b<, rjtemp = 8rj1, rj2, rj3, rj4<;

rktemp = : >;
flat@siD flat@sxD flat@syD flat@szD
, , ,

S = 8si, sx, sy, sz<; rjmtemp = 8rjm1, rjm2, rjm3, rjm4<;

2 2 2 2

l = flat@Table@Conjugate@rktemp@@kDDD.rjmtemp@@jDD, 8j, 1, 4<, 8k, 1, 4<DD;

b = Table@flat@Table@Conjugate@rktemp@@Floor@Hp - 1L ê 4 + 1DDDD.
flat@S@@mDD.rjtemp@@Mod@p - 1, 4D + 1DD.S@@nDDD, 8m, 1, 4<, 8n, 1, 4<DD,
8p, 1, 16<D; unflat@Chop@Inverse@bD.lD, 4DF

ChiT@t1_, t2_, t3_, t4_, t5_, t6_, t7_,

Simplify@dagger@88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0, 0<, 8t11 + Â t12, t7 + Â t8, t3, 0<,
t8_, t9_, t10_, t11_, t12_, t13_, t14_, t15_, t16_D :=

8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<<D.88t1, 0, 0, 0<, 8t5 + Â t6, t2, 0, 0<,
8t11 + Â t12, t7 + Â t8, t3, 0<, 8t15 + Â t16, t13 + Â t14, t9 + Â t10, t4<<,
Assumptions Ø 8 8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10,
t11, t12, t13, t14, t15, t16< œ Reals<D
Clear@lD
 b. mathematica code: tomography 295
ProcessMin2@t1_, t2_, t3_, t4_, t5_, t6_, t7_, t8_, t9_, t10_, t11_, t12_,
t13_, t14_, t15_, t16_, p1m1_, p1m2_, p1m3_, p1m4_, p2m1_, p2m2_, p2m3_,
p2m4_, p3m1_, p3m2_, p3m3_, p3m4_, p4m1_, p4m2_, p4m3_, p4m4_, l_D :=

J4 I-p1m1 + t12 + t112 + t122 + t162 + Ht15 - t4L2 + t52 + t62 M +

1 2

4
4 I-p2m2 + t12 + t112 + t122 + t162 + Ht15 + t4L2 + t52 + t62 M +
2

4 I-p4m4 + t12 + Ht13 - t15L2 + Ht14 - t16L2 + Ht2 - t5L2 + t62 + Ht11 - t7L2 +
Ht12 - t8L2 M + 4 I-p2m1 + Ht10 + t13L2 + t22 + t72 + Ht3 - t8L2 + Ht14 - t9L2 M +
2 2

I-2 p3m1 + t12 + Ht10 + t13 + t16L2 + t52 + Ht2 + t6L2 + Ht12 + t7L2 +
Ht11 + t3 - t8L2 + Ht14 - t15 + t4 - t9L2 M +
2

4 I-p1m2 + Ht10 - t13L2 + t22 + t72 + Ht3 + t8L2 + Ht14 + t9L2 M +

2

4 I-p3m3 + t12 + t122 + Ht10 + t16L2 + Ht11 + t3L2 + t52 + t62 + Ht15 + t9L2 M +
2

I-2 p3m4 + t12 + Ht10 - t14 + t16 - t4L2 + Ht2 - t5L2 + t62 +
Ht11 + t3 - t7L2 + Ht12 - t8L2 + H-t13 + t15 + t9L2 M +
2

I-2 p4m3 + t12 + Ht10 - t14 + t16 + t4L2 + Ht2 - t5L2 + t62 + Ht11 + t3 - t7L2 +
Ht12 - t8L2 + H-t13 + t15 + t9L2 M + I-2 p1m4 + t12 + Ht10 - t13 + t15 - t4L2 +
2

Ht2 - t5L2 + t62 + Ht11 - t7L2 + H-t12 + t3 + t8L2 + Ht14 - t16 + t9L2 M +
2

I-2 p4m2 + t12 + Ht10 - t13 + t15 + t4L2 + Ht2 - t5L2 + t62 + Ht11 - t7L2 +
H-t12 + t3 + t8L2 + Ht14 - t16 + t9L2 M + I-2 p2m4 + t12 + Ht10 + t13 - t15 - t4L2 +
2

Ht2 - t5L2 + t62 + Ht11 - t7L2 + Ht12 + t3 - t8L2 + H-t14 + t16 + t9L2 M +
2

I-2 p4m1 + t12 + Ht10 + t13 - t15 + t4L2 + Ht2 - t5L2 + t62 + Ht11 - t7L2 +
Ht12 + t3 - t8L2 + H-t14 + t16 + t9L2 M + I-2 p1m3 + t12 + Ht10 - t13 + t16L2 +
2

t52 + Ht2 - t6L2 + Ht12 - t7L2 + Ht11 + t3 + t8L2 + Ht14 + t15 - t4 + t9L2 M +
2

I-2 p2m3 + t12 + Ht10 + t13 + t16L2 + t52 + Ht2 + t6L2 + Ht12 + t7L2 +
Ht11 + t3 - t8L2 + H-t14 + t15 + t4 + t9L2 M +
2

I-2 p3m2 + t12 + Ht10 - t13 + t16L2 + t52 + Ht2 - t6L2 + Ht12 - t7L2 +

Ht11 + t3 + t8L2 + Ht14 + t15 + t4 + t9L2 M N +

2

l 4 J4 Ht13 t15 + t14 t16 - t10 t4 + t2 t5 + t11 t7 + t12 t8L2 +

4 Ht10 t13 + t15 t4 - t3 t8 - t14 t9L2 + 4 Ht10 t16 + t11 t3 + t14 t4 + t15 t9L2 +
I-1 + t12 + t102 + t112 + t122 + t132 + t142 + t152 +

t162 + t22 + t32 + t42 + t52 + t62 + t72 + t82 + t92 M N

2

MLEP2@p1m1_, p1m2_, p1m3_, p1m4_, p2m1_, p2m2_, p2m3_, p2m4_,

8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16, sol<,
p3m1_, p3m2_, p3m3_, p3m4_, p4m1_, p4m2_, p4m3_, p4m4_, l_D := Module@

sol = NMinimize@ProcessMin2@t1, t2, t3, t4, t5, t6, t7, t8, t9, t10,
t11, t12, t13, t14, t15, t16, p1m1, p1m2, p1m3, p1m4, p2m1, p2m2,

8t1, t2, t3, t4, t5, t6, t7, t8, t9, t10, t11, t12, t13, t14, t15, t16<,
p2m3, p2m4, p3m1, p3m2, p3m3, p3m4, p4m1, p4m2, p4m3, p4m4, lD,

t8, t9, t10, t11, t12, t13, t14, t15, t16D ê. sol@@2DDD
MaxIterations Ø 10 000D; ChiT@t1, t2, t3, t4, t5, t6, t7,
 Copyright Permissions

• Figures 6.2 to 6.4 reproduced with permission from :

Lev S. Bishop, J. M. Chow, Jens Koch, A. A. Houck, M. H. Devoret, E. Thuneberg,
S. M. Girvin, and R. J. Schoelkopf, Nature Phys. 5, 105–109 (2009).
Copyright © 2009, Nature Publishing Group.

• Figure 5.4 reproduced with permission from:

Jens Koch, Terri M. Yu, Jay Gambetta, A. A. Houck, D. I. Schuster, J. Majer, Alexan-
dre Blais, M. H. Devoret, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. A 76, 042319
(2007).
Copyright (2007) by the American Physical Society.

• Figure 4.2 reproduced with permission from :

F. Motzoi, J. M. Gambetta, P. Rebentrost, and F. K. Wilhelm, Phys. Rev. Lett. 103,
110501 (2009).
Copyright (2009) by the American Physical Society.

• Figures 6.8, 6.10, 6.11, 6.13 and 6.15 reproduced with permission from :
J. M. Chow, J. M. Gambetta, L. Tornberg, Jens Koch, Lev S. Bishop, A. A. Houck,
B. R. Johnson, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Phys. Rev. Lett. 102,
090502 (2009).
Copyright (2009) by the American Physical Society.

• Figures 7.1, 7.4 to 7.8, 7.10 and 7.11 reproduced with permission from :
J. Majer, J. M. Chow, J. M. Gambetta, Jens Koch, B. R. Johnson, J. A. Schreier, L. Frun-
zio, D. I. Schuster, A. A. Houck, A. Wallraff, A. Blais, M. H. Devoret, S. M. Girvin,
and R. J. Schoelkopf, Nature 449, 443–447 (2007).
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• Figures 8.1, 8.2, 8.4, 8.5, 8.11, 9.2, 9.4 and 9.5 reproduced with permission from :
L. DiCarlo, J. M. Chow, J. M. Gambetta, Lev S. Bishop, B. R. Johnson, D. I. Schuster,
J. Majer, A. Blais, L. Frunzio, S. M. Girvin, and R. J. Schoelkopf, Nature 460, 240–
244 (2009).
Copyright © 2009, Nature Publishing Group.