Letter doi:10.

1038/nature23879

Hardware-efficient variational quantum eigensolver

for small molecules and quantum magnets
Abhinav Kandala1*, Antonio Mezzacapo1*, Kristan Temme1, Maika Takita1, Markus Brink1, Jerry M. Chow1 & Jay M. Gambetta1

Quantum computers can be used to address electronic-structure ­ roblem using the quantum phase estimation algorithm15. Although
p
problems and problems in materials science and condensed matter this ­algorithm can produce extremely accurate energy estimates for
physics that can be formulated as interacting fermionic problems, ­quantum chemistry2,3,5,8, it applies stringent requirements on the
problems which stretch the limits of existing high-performance ­coherence of the quantum hardware.
computers1. Finding exact solutions to such problems numerically An alternative approach is to use quantum optimizers, which
has a computational cost that scales exponentially with the size of have previously demonstrated utility, for example, for combinatorial
the system, and Monte Carlo methods are unsuitable owing to the optimization problems16,17 and in quantum chemistry as variational
fermionic sign problem. These limitations of classical computational ­quantum eigensolvers (VQEs) where they were introduced to reduce
methods have made solving even few-atom electronic-structure the ­coherence requirements on quantum hardware4,18,19. The VQE uses
problems interesting for implementation using medium-sized Ritz’s variational principle to prepare approximations to the ground
quantum computers. Yet experimental implementations have state and its energy. In this approach, the quantum computer is used to
so far been restricted to molecules involving only hydrogen and prepare variational trial states that depend on a set of parameters. The
helium2–8. Here we demonstrate the experimental optimization expectation value of the energy is then estimated and used in a classical
of Hamiltonian problems with up to six qubits and more than one optimizer to generate a new set of improved parameters. The advantage
hundred Pauli terms, determining the ground-state energy for of a VQE over classical simulation methods is that it can prepare trial
molecules of increasing size, up to BeH2. We achieve this result by states that are not amenable to efficient classical numerics.
using a variational quantum eigenvalue solver (eigensolver) with The VQE approach realized in experiments has so far been limited
efficiently prepared trial states that are tailored specifically to the by different factors. Typically, a unitary coupled cluster ansatz for the
interactions that are available in our quantum processor, combined trial state is considered6,7, which has a number of parameters that
with a compact encoding of fermionic Hamiltonians9 and a robust scales quartically with the number of spin orbitals that are considered
stochastic optimization routine10. We demonstrate the flexibility of in the single- and double-excitation approximation. Furthermore,
our approach by applying it to a problem of quantum magnetism, when implementing the unitary coupled cluster ansatz on a quantum
an antiferromagnetic Heisenberg model in an external magnetic computer, Trotterization errors need to be accounted for19–21. Here we
field. In all cases, we find agreement between our experiments and introduce and implement a hardware-efficient ansatz preparation for a
numerical simulations using a model of the device with noise. Our VQE, whereby trial states are parameterized by quantum gates that are
results help to elucidate the requirements for scaling the method tailored to the physical device that is available. We show numerically the
to larger systems and for bridging the gap between key problems viability of such trial states for small electronic-structure problems and
in high-performance computing and their implementation on use a superconducting quantum processor to perform optimizations of
quantum hardware. the molecular energies of H2, LiH and BeH2, and extend its application
The fundamental goal in electronic-structure problems is to solve to a Heisenberg antiferromagnetic model in an external magnetic field.
for the ground-state energy of many-body interacting fermionic The device used in the experiments is a superconducting quantum
Hamiltonians. Solving this problem on a quantum computer relies on processor with six fixed-frequency transmon qubits, together with a
a mapping between fermionic and qubit operators11, which restates central weakly tunable asymmetric transmon qubit22. The device is
the problem as a specific instance of a local Hamiltonian problem on a cooled in a dilution refrigerator, where it is thermally anchored to its
set of qubits. Given a k-local Hamiltonian H, composed of terms that mixing chamber plate at 25 mK. The experiments discussed here make
act on at most k qubits, the solution to the local Hamiltonian problem use of six of these qubits (labelled Q1–Q6; Fig. 1b). The qubits are
amounts to finding its ground-state eigenvalue EG and ground state coupled via two superconducting coplanar waveguide resonators, and
|​ΦG〉​, which satisfy can be controlled individually and read out using independent read-
out resonators.
H Φ G〉 = E G Φ G〉 The hardware-efficient trial states that we consider use the naturally
available entangling interactions of the superconducting hardware,
So far, no efficient algorithm is known that can solve this problem in which are described by a drift Hamiltonian H0. This Hamiltonian
its fully general form. For k ≥​ 2, the problem is known to be quantum ­generates the entanglers, which are unitary operators of the form
Merlin Arthur (QMA)-complete12; however, it is expected that physical UENT =​  exp(−​iH0τ), which entangle all the qubits in the circuit, where
systems have Hamiltonians that can be solved efficiently on a quantum τ is the evolution time. These entanglers are interleaved with a­ rbitrary
computer, while remaining hard to solve on a classical computer. single-qubit Euler rotations, which are implemented as a combination
Following Feynman’s idea for quantum simulation, a quantum of Z and X gates, U q ,i (θ ) = Z qq, i X qq, i Z qq, i, where θ ­represents the Euler
θ1 θ2 θ3
­algorithm for the ground-state problem of interacting fermions has
been proposed13,14. The approach relies on a ‘good’ initial state— angles, q identifies the qubit and i =​  0, 1, …, d refers to the depth posi-
one that has a large overlap with the ground state—and solves the tion, as depicted in Fig. 1c. The N-qubit trial states are obtained from

1
IBM T.J. Watson Research Center, Yorktown Heights, New York 10598, USA.
*These authors contributed equally to this work.

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 Letter RESEARCH

a Q1 c
Q2
Q3 I X–π/2–
Q1 |0〉 U1,0(Tk) U1,1(Tk)
Q7 Yπ/2
Q4
I X–π/2
Q5 Q2 |0〉 U2,0(Tk) U2,1(Tk) Yπ/2
Q6
Q8 I X–π/2
Q3 |0〉 U3,0(Tk) U3,1(Tk)
Yπ/2

UENT
I X–π/2
2s 2px 1s 1s′ 2s 2px 1s 1s′ Q4 |0〉 U4,0(Tk) U4,1(Tk)
Yπ/2
b
I X–π/2
Q5 |0〉 U5,0(Tk) U5,1(Tk) Yπ/2

I X–π/2
Q6 |0〉 U6,0(Tk) U6,1(Tk) Yπ/2
d
d
Q6 Q4 Q1 Q2
Q1 |0〉
Q5 Q3 Q2 |0〉
Q3 |0〉
UENT
Q4 |0〉
Q5 |0〉
Q6 |0〉
1 mm
d

Figure 1 | Quantum chemistry on a superconducting quantum for control and read-out. c, Hardware-efficient quantum circuit for trial-
processor. Solving electronic-structure problems on a quantum computer state preparation and energy estimation, shown here for six qubits. For
relies on mappings between fermionic and qubit operators. a, Parity each iteration k, the circuit is composed of a sequence of interleaved
mapping of eight spin orbitals (drawn in blue and red, not to scale) onto single-qubit rotations Uq,d(θk) and entangling unitary operations UENT
eight qubits, which are then reduced to six qubits owing to fermionic that entangle all of the qubits in the circuit. A final set of post-rotations
spin and parity symmetries. The length of the bars indicate the parity of (I, X−π/2 or Yπ/2) before the qubits are read out is used to measure the
the spin orbitals that are encoded in each qubit. b, False-coloured optical expectation values of the individual Pauli terms in the Hamiltonian and to
micrograph of the superconducting quantum processor with seven estimate the energy of the trial state. d, An example of the pulse sequence
transmon qubits. These qubits are coupled via two coplanar waveguide for the preparation of a six-qubit trial state, in which UENT is implemented
resonators (violet) and have individual coplanar waveguide resonators as a sequence of two-qubit cross-resonance gates.

the state |​00…0〉​, applying d entanglers UENT that ­alternate with N Euler maximal entanglement23. We set our two-qubit gate times at 150 ns, to
rotations, giving try to minimize the effect of decoherence without compromising the
N N accuracy of the optimization outcome; see Supplementary Information.
|Φ(θ )〉 = ∏ [U (θ )] × UENT × ∏ [U
q , d q , d − 1(θ )] ×  × UENT × After each trial state is prepared, we estimate the associated energy
q =1 q =1 by measuring the expectation values of the individual Pauli terms in
N the Hamiltonian. These estimates are affected by stochastic fluctua-
∏ [U q ,0(θ )]|00 … 0〉 tions due to finite sampling. Different post-rotations are applied after
q =1 trial-state preparation for sampling different Pauli operators (Fig. 1c, d).
We group the Pauli operators into tensor product basis sets that require
Because the qubits are all initialized in their ground state |​0〉​, the first the same post-rotations. We numerically show that such grouping
set of Z rotations of Uq,0(θ) is not implemented, resulting in a total of reduces the energy fluctuations, while keeping the same total number
p =​  N(3d +​2) independent angles. In the experiment, the evolution of samples, thereby reducing the time overhead for energy estimation;
time τ and the individual couplings in H0 can be controlled. However, see Supplementary Information. The energy estimates are then used
numerical simulations indicate that accurate optimizations are in a gradient descent algorithm that relies on a simultaneous perturba-
obtained for fixed-phase entanglers UENT, leaving the p control angles tion stochastic approximation (SPSA) to update the control parameters.
as v­ ariational parameters. Our hardware-efficient approach does not The SPSA algorithm approximates the gradient using only two energy
rely on the accurate implementation of specific two-qubit gates and can measurements, regardless of the dimensions of the parameter space p,
be used with any UENT that generates sufficient entanglement. This is achieving a level of accuracy comparable to that of standard gradient
in ­contrast to unitary coupled-cluster trial states, which require high-­ descent methods, in the presence of stochastic fluctuations10. This is
fidelity ­quantum gates that approximate a unitary operator tailored on crucial for optimizing over many qubits and long depths for trial-state
the basis of a theoretical ansatz. For the experiments considered here, preparation, enabling us to optimize over a number of parameters as
the entanglers UENT are composed of a sequence of two-qubit cross-­ large as p =​  30.
resonance gates23. Simulations as a function of entangler phase show To address molecular problems on our quantum processor, we rely on
plateaus of minimal energy error around gate phases that correspond a compact encoding of the second-quantized fermionic Hamiltonians
to the maximal pairwise concurrence; see Supplementary Information. onto qubits. The Hamiltonian for molecular H2 has four spin orbitals,
We therefore set the entangler evolution time τ at the beginning of such representing the spin-degenerate 1s orbitals of the two hydrogen atoms.
plateaus, to reduce decoherence effects. We use a binary tree encoding11 to map the Hamiltonian to a four-
In our experiments, the Z rotations are implemented as frame qubit system, and remove the two qubits that are associated with the
changes in the control software24, whereas the X rotations are imple- spin parities of the system9. The Hamiltonian for BeH2 is defined on
mented by appropriately scaling the amplitude of calibrated Xp pulses, the basis of the 1s, 2s and 2px orbitals that are associated with Be, and
using a fixed total time of 100 ns for every single-qubit rotation. The the 1s orbital that is associated with each H atom, for a total of ten spin
cross-resonance gates that compose UENT are implemented by driving orbitals. We then assume perfect filling of the innermost two 1s spin
a control qubit Qc with a microwave pulse that is resonant with a target orbitals of Be, after shifting their energies by diagonalizing the non-
qubit Qt. We use Hamiltonian tomography of these gates to determine interacting part of the fermionic Hamiltonian. We map the eight-
the strengths of the various interaction terms, and the gate time for spin-orbital Hamiltonian of BeH2 using parity mapping and, as in

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 RESEARCH Letter

–13.6 q ' q q
q
X T q,1,± q
X T q,0,± Z T q,0,± Z T q,1,± Z T q,1,±
π 2 3 1
2
3

–13.8
π/2

(rad)
–14.0
0

Tq,i,±
j
–14.2
–π/2
Energy (hartree)

–14.4
–π
–14.6
0 200 0 200 0 200 0 200 0 200

–14.8

–15.0

–15.2

Final experimental result

–15.4
Exact

–15.6
0 50 100 150 200 250
Iteration, k
Figure 2 | Experimental implementation of six-qubit optimization. The optimization of 30 Euler angles that control the trial state preparation.
minimum energy of the six-qubit Hamiltonian describing BeH2 with an Each colour refers to a particular qubit (Q1–Q6; q =​ 1, 2, …), following the
interatomic distance of l =​ 1.7 Å (data points) is plotted along with the colour scheme in Fig. 1. The final energy estimate (green dashed line) is
exact value (black dashed line). For each iteration k, the gradient at each obtained using the average angle over the last 25 angle updates (indicated
control θk is approximated using 1,000 samples for energy estimation by the green dotted arrow), to mitigate the effect of stochastic fluctuations,
at θ+ −
k (blue) and θ k (red), which are perturbations to θk along opposite and with a higher number of samples (100,000), to obtain a more accurate
directions of a random axis in parameter space. The error bars correspond energy estimation.
to the standard error of the mean. The inset shows the simultaneous

the case of H2, remove two qubits associated with the spin–parity The results from an optimization procedure are illustrated in Fig. 2,
s­ ymmetries, reducing the Hamiltonian to a six-qubit problem that using the Hamiltonian for BeH2 at the interatomic distance of 1.7 Å.
encodes eight spin orbitals. A similar approach is used to map LiH Although using a large number of entanglers UENT helps to achieve
onto four qubits. The Hamiltonians for H2, LiH and BeH2 at their better energy estimates in the absence of noise, the combined effect
­lowest-energy interatomic distances (bond distance) are given e­ xplicitly of decoherence and finite sampling sets the optimal depth for opti-
in Supplementary Information. mizations on our quantum hardware to 0–2 entanglers. The results

a b c
0.4 –6.6 –12.0
100 40
40
Q6 Q1 Q6 Q1 Q6 Q1
0.2 –6.8 –12.5
C
C

50 20
5

R1
R1

6–

20
R

–3
–3

0 Q5 Q4 Q3 Q5 Q4 Q3 Q5 CR4–5 Q4 Q3
CR2–1
CR2–1

0 –7.0 0 –13.0 0
C

C
C

R2

R2
R2
Energy (hartree)

–0.2
–4

–4
–4

Q7 Q2 Q7 Q2 –13.5 Q7 Q2
–7.2
H H Be
–0.4 H H
–14.0
–7.4 Li H
–0.6
–14.5
–7.6
–0.8
–15.0
–1.0 –7.8
–15.5
–1.2 –8.0
0 1 2 3 4 1 2 3 4 5 1 2 3 4 5
Interatomic distance (Å) Interatomic distance (Å) Interatomic distance (Å)
Figure 3 | Application to quantum chemistry. a–c, Experimental results outcomes at each interatomic distance. The top insets in each panel
(black filled circles), exact energy surfaces (dotted lines) and density plots highlight the qubits used for the experiment and the cross-resonance
(shading; see colour scales) of outcomes from numerical simulations, gates (arrows, labelled CRc–t; where ‘c’ denotes the control qubit and ‘t’ the
for several interatomic distances for H2 (a), LiH (b) and BeH2 (c). The target qubit) that constitute UENT. The bottom insets are representations
experimental and numerical results presented are for circuits of depth of the molecular geometry (not to scale). For all the three molecules,
d =​ 1. The error bars on the experimental data are smaller than the the deviation of the experimental results from the exact curves is well
size of the markers. The density plots are obtained from 100 numerical explained by the stochastic simulations.

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 Letter RESEARCH

a b Exact d=0 d=2 c

4
Q6 30
Q1 0
2 –2 50

Magnetization (a.u.)
20
Q5 Q4 Q3 10

Energy (a.u.)
0 –1
Energy (a.u.)

0 0
–2 –4
Q7 Q2 50 –2
–4
20 –3
–6
–6
0 0
–8 –4
–8
0 50 100 150 200 250 0 0.2 0.4 0.6 0.8 1 0 0.2 0.4 0.6 0.8 1
Iteration, k J/B J/B
Figure 4 | Application to quantum magnetism. The optimization highlights the qubits used for the experiment and the cross-resonance
of a four-qubit Heisenberg model on a square lattice, in an external gates (arrows) that constitute UENT. b, c, Experimental results for d =​  0
magnetic field, using circuits of depth d =​ 0 (blue) and d =​ 2 (red) for state (blue squares) and d =​ 2 (red circles) plotted along with exact values (black
preparation is compared. a, Energy optimization for J/B =​ 1 (data points, dashed lines) and density plots of 100 numerical outcomes, for energy (b)
with dashed lines indicating the final energy estimate, determined as in and magnetization (c), for a range of values of J/B.
Fig. 2), plotted along with the exact energy (dashed black line). The inset

presented in Fig. 2 were obtained using a circuit of depth d =​ 1, with a H = J ∑ 〈ij 〉 (X i X j + YiYj + Z iZ j ) + B ∑ i Z i
total of 30 Euler control angles associated with six qubits. The inset of
Fig. 2 shows the simultaneous perturbation of 30 Euler angles as the
where 〈​ij〉​indicates the nearest-neighbour pairs, J is the strength of the
energy estimates are updated.
spin–spin interaction and B the magnetic field along the Z direction.
To obtain the potential-energy surfaces for H2, LiH and BeH2, we
Similar spin models have previously been simulated using trapped
search for the ground-state energy of their molecular Hamiltonians,
ions28. We use our technique to solve for the ground-state energy of
using two, four and six qubits, respectively, a depth d =​ 1 and a range
the system for a range of J/B values. When J =​ 0, the ground state is
of different interatomic distances. The experimental results are com-
completely separable and the best estimates are obtained for a depth
pared with the ground-state energies obtained from exact diagonaliza-
of d =​  0. As J is increased, the ground state is increasingly entangled
tion and outcomes from numerical simulations in Fig. 3. The coloured
and the best estimates are instead obtained for d =​ 2, despite the
density plots in each panel are obtained from 100 numerical optimiza-
increased d ­ ecoherence that is caused by using two entanglers for
tions for each interatomic distance, using cross-resonance entangling
­ reparation. This behaviour is shown in Fig. 4a for J/B =​  1.
­trial-state p
gates with the same topology as in the experiments. These numerics
The e­ xperimental results are compared with the exact ground-state
account for decoherence effects, which are simulated by adding ampli-
energies for a range of J/B values in Fig. 4b, and our deviations are
tude damping and dephasing channels after each layer of quantum
captured by the density plots of the numerical outcomes that account
gates. The effect of finite sampling on the optimization algorithm is
for noisy energy estimations and decoherence. Furthermore, in Fig. 4c,
taken into account by numerically sampling the individual Pauli terms
we show that our approach can be used to evaluate observables such as
in the Hamiltonian, and adding their averages. The strengths of the
the magnetization of the system Mz.
noise channels are derived from the measured values of the coherence
The experiments presented here demonstrate that a hardware-­
times T1 and T ∗2 . In addition to the effects of decoherence and noisy
efficient VQE implemented on a six-qubit superconducting quantum
energy estimates, the deviations are also due to low circuit depth for
processor is capable of addressing molecular problems beyond period I
trial-state preparation, which, for example, explains the kink in the
elements, up to BeH2. A numerical investigation of the h ­ ardware
range l =​ 2.5–3 Å in Fig. 3b. In the absence of noise, critical depths of
involved suggests that substantial improvements in ­coherence
d =​  1, d =​  8 and d =​ 28 are required to achieve chemical accuracy (an
and s­ ampling are needed to improve the accuracy of a VQE for
energy error of approximately 0.0016 hartree) on the current experi-
the m­ olecules that we addressed; see Supplementary Information. For
mental connectivities for H2, LiH and BeH2, respectively; critical depths
more complex problems, increased coherence and faster gates would
of d =​  1, d =​  6 and d =​ 16 are required to achieve chemical accuracy on
enable larger circuit depths for state preparation, whereas increased
the respective all-to-all connectivities; see Supplementary Information.
on-chip qubit connectivity is crucial for reducing the ­critical depth
By contrast, a generic unitary coupled-cluster ansatz truncated to
requirements to achieve chemical accuracy. The use of fast reset
second order for an eight-orbital molecule such as our model of BeH2
schemes29 would enable increased sampling rates, improving the
would require 4,160 fermionic variational terms, which, after account-
effectiveness of the classical optimizer and reducing time overheads.
ing for fermionic mappings and Trotterization, would generate a num-
The performance of the quantum–classical optimization could be
ber of quantum gates that is of the same order. The scaling of resources
further improved by using variants30 of the SPSA protocol. Trial-
and the noise requirements necessary for achieving chemical accuracy
state-preparation circuits, which combine ansatzes from classical
using ­hardware-efficient trial states are detailed in Supplementary
approximation methods and hardware-efficient gates, could be inves-
Information. We emphasize that our approach is unaffected by coherent
tigated further to improve on the current state ansatz. Finally, even
gate errors, which shifts the focus to reducing incoherent errors,
before the advent of fault-tolerant architechtures, the agreement of our
­favouring our fixed-frequency, all-microwave-controlled qubit
experimental results with the noise models that we considered opens
architecture. Furthermore, the effect of incoherent errors can be
up a path to error-mitigation protocols for experimentally accessible
­mitigated as recently proposed25–27, without requiring additional
circuit depths25–27.
­quantum resources.
We now demonstrate the applicability of our technique to a problem Data Availability The data that support the findings of this study are available
of quantum magnetism, and show that, with the same noisy quantum from the corresponding author on reasonable request.
hardware, the advantage of using greater circuits depths is crucially
dependent on the target Hamiltonian. Specifically, we consider a received 13 April; accepted 18 July 2017.
four-qubit Heisenberg model on a square lattice, in the presence
1. National Energy Research Scientific Computing Center 2015 Annual Report
of an external magnetic field. The model is described by the http://www.nersc.gov/assets/Annual-Reports/2015NERSCAnnualReportFinal.
Hamiltonian pdf (2015).

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 RESEARCH Letter

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