Multi-QIDA method for VQE state preparation in molecular

systems
Fabio Tarocco1,2 , Davide Materia1,2 , Leonardo Ratini1,2 , and Leonardo Guidoni2, :
arXiv:2508.11270v1 [quant-ph] 15 Aug 2025

1
Dipartimento di Ingegneria e Scienze dell’Informazione e Matematica, Università degli
Studi dell’Aquila, Coppito, L’Aquila, Italy
2
Dipartimento di Scienze Fisiche e Chimiche, Università degli Studi dell’Aquila, Coppito,
L’Aquila, Italy
:
Email: leonardo.guidoni@univaq.it

August 18, 2025

Abstract
The development of quantum algorithms and their application to quantum chemistry has introduced
new opportunities for solving complex molecular problems that are computationally infeasible for classical
methods. In quantum chemistry, the Variational Quantum Eigensolver (VQE) is a hybrid quantum-classical
algorithm designed to estimate ground-state energies of molecular systems. Despite its promise, VQE faces
challenges such as scalability issues, high circuit depths, and barren plateaus that make the optimization
of the variational wavefunction. To mitigate these challenges, the Quantum Information Driven Ansatz
(QIDA) leverages Quantum Mutual Information (QMI) to construct compact, correlation-driven circuits.
In this work, we go back to the original field of application of QIDA, by applying the already defined
Multi-Threshold Quantum Information Driven Ansatz (Multi-QIDA) methodology on Molecular Systems.
to systematically construct shallow, layered quantum circuits starting from approximate QMI matrices
obtained by Quantum Chemistry calculations. The Multi-QIDA approach combines efficient creation of
the QMI map, reduction of the number of correlators required by exploiting Minimum/Maximum spanning
tress, and an iterative layer-wise VQE optimization routine. These enhancements allow the method to
recover missing correlations in molecular systems while maintaining computational efficiency. Additionally,
the approach incorporates alternative gate constructions, such as SO(4) correlators, to enhance the circuit
expressibility without significantly increasing the circuit complexity.
We benchmark Multi-QIDA on systems ranging from small molecules like H2 O, BeH2 , and NH3 in
Iterative Natural Orbitals (INOs) basis set, to active-space models such as H2 O-6-31G-CAS(4,4) and N2 -cc-
pVTZ-CAS(6,6), comparing it to traditional hardware-efficient ansätze. The results show that Multi-QIDA
consistently outperforms the ladder topology ans”atze in terms of energy accuracy, correlation recovery,
and resource utilization. Interestingly, in addition to the higher fidelity to the exact ground state, we
observe a significant improvement of the quality of the variational wavefunction which better preserves the
correct symmetries, such as Sˆz , Sˆ2 , and N̂ .

1
1 Introduction the ansätze, increasing complexity in parameterized
quantum circuits (PQC) results in longer and deeper
The Variational Quantum Eigensolver (VQE) circuits, which not only heightens the risk of error
method utilizes a hybrid quantum-classical approach accumulation but also contributes to the emergence
to estimate ground state energies of molecular of Barren Plateaus [21], an issue characterized by an
systems by optimizing a parameterized quantum exponentially flat optimization landscape. While ad-
ansatz through iterative energy minimization. Since justments in optimization techniques or error mitiga-
about a decade the method has been largely studied tion strategies do not directly resolve barren plateaus,
(and criticized) for its employnment into near-term employing shallow and adaptively structured ansätze
applications designed for quantum chemistry [1–7]. has proven to be an effective countermeasure. Exam-
Its simplicity instead hides scalability challenges ples of such approaches include ADAPT-VQE, a vari-
because of to the growth of parameter space, the flat- ant of the UCC method, which builds a wavefunction
ness of the corresponding energy landscape, the in- only by exploiting energetically meaningful single and
tensive exchange of data between quantum and clas- double excitations that are applied on the circuit in
sical devices, and circuit depth on noisy quantum de- an adaptive way [9, 10, 15]. Another approach is to
vices which need to be mitigated. To address some of optimize both the Hamiltonian and the wavefunction
these issues, various improved algorithms have been with the objective of compacting the circuit and re-
proposed [8–11]. ducing the depth, done by applying the WAHTOR
In this respect, a fundamental role is led by the algorithm [22]. Shallow circuits that can also encode
choice of the parametrized wavefunction that is go- chemically relevant information and being build, for
ing to be used as a trial wavefunction together with example, following the Mutual Information present
the VQE procedure. The shape and the structure in the system, as done with Quantum Information
of the ansatz are non-trivial and system-dependent, Driven Ansatz [23]. This approach allows to de-
and in general, two types of ansätze can be defined. fine quickly initial guess with a limited number of
The first class exploits wavefunctions directly con- CNOTs. Obtaining energetically correct results with
structed to leverage the characteristics of quantum a good overlap with the ground state is not only use-
hardware. This empirical approach, known as the ful as a starting guess for more complex ansatz con-
Heuristic Ansatz [3, 12–15], comprises repetitions of struction protocol or other VQE-based procedures,
blocks of parametrized rotations and entanglement but also as a trial wavefunction from which samples
gates. It is designed without relying on information can be extracted. Recently, different works in which a
about the physical system, focusing solely on exploit- non-VQE-based, mainly exploiting the quantum ver-
ing the quantum hardware’s capabilities. While the sion of a Selected Configuration Interaction (SCI) ap-
Heuristic Ansatz better utilizes the quantum hard- proach, called Quantum SCI (QSCI) [24], Quantum
ware, it comes at the cost of losing the physical Subspace Diagonalization (QSD) [25], or Sample-
meaning associated with the wavefunction interpreta- Based Quantum Diagonalization (SQD) [26], have
tion, with the advantage of considering significantly been proving relevant industrial results and applica-
shallower circuits, providing a potential avenue to tion of Quantum Algorithm to quantum utility-scale
address scalability concerns. In contrast, the sec- [26, 27]. In these works, the principal actor is the
ond family of ansätze involves translating classical trial wavefunction from which the samples are drawn,
Quantum Chemistry methods into the language of which, for example, can be either a fixed Quantum
quantum computation. [16–20]. One of the most Number Preserving (QNP) circuit [28] or a chemi-
important chemically inspired ansatz is the Unitary cally inspired hardware-efficient Local Unitary Cou-
Coupled Cluster (UCC) which directly translated the pled Jastrow (LUCJ) [26]. Despite these advantages,
wavefunction structure as defined for the ”classical” being able to construct compact wavefunctions that
Coupled cluster theory. [16–20] can build effective trial wavefunctions that can be
Regardless of the approach used to construct used to sample relevant determinants with the cor-

2
rect quantum symmetries is still a challenge. gates used in this work, the SO4 gates, in Section
With this work, we wanted to test the performance 3.1, while the selection criteria based on spanning
of the Multi-QIDA approach introduced by us for tree to place them follows in Section 3.2. Thanks to
lattice spin-systems [29] on molecular systems. In the selected pairs, the ansatz is then built following
particular, we combined the idea behind the usage the procedure explained in Section 3.3 and optimized
of Natural Orbitals to obtain a more correlated and accordingly to the optimization scheme of Section 3.4.
compact wavefunction and the application of multi- We then presented the system that have been studied
ple layers of QIDA-based circuits. We also introduced in this work, both simple molecule and active-space
two different ways of reducing the number of corre- selected systems, are presented in Section 4.1, fol-
lators to insert in each QIDA-layer, namely, maximal lowed by the metrics used to evaluate the efficiency
correlation and topological distance, based on Span- of the ansätze and the symmetry measures computed,
ning Trees. The approach has been integrated with in Section 4.3. The simulation details are explained
the QMI builder tool SparQ [30] to efficiently com- in Section 4.4. Finally, in Section 5, results are shown
pute reference Quantum Mutual Information matri- and explained.
ces. With this new method, we have been able to
partially recover the missing electron correlations by
building and optimizing a layered-structured ansätz 2 Background
which is extended at each iteration by including a
In this section, we collected very briefly all the meth-
newly computed QIDA-layer based on the QMI. We
ods and related principles that have been used in our
have thus amplified the results obtained by the stand-
work.
alone QIDA ansatz in its same field of application
by recovering mid/low-correlated qubit-pairs. In this
way, we have been able to obtain shallow circuits 2.1 Eigenproblem for Quantum
that, compared to the same CNOT-count of a HEA Chemistry Hamiltonian
ladder-fashion circuit, reach higher correlation energy
Given a molecule and a set of spin molecular orbitals
with increased precision. Anticipating our results,
(MOs), the second-quantized electronic Hamiltonian
the optimised ansätze turned out are also to be qual-
is defined as follows
itatively better than simpler ones in terms of preserv-
ing the relevant spin and electronic symmetries and
ÿ 1 ÿ
Ĥ “ hij â:i âj ` Γijkl â:i â:j âk âl , (1)
quantities of the exact ground states, like Ŝz , Ŝ 2 , and ij
2 ijkl
N̂e .
This paper is split into the following sections. where a:i and ai are creation and annihilation oper-
Firstly, with Section 2, we provide a brief introduc- ators, respectively, while hij and Γijkl are the one-
tion on the theoretical background required for the body and two-body integrals. In the summation,
full explanation of the Multi-QIDA approach. In the indices of Equation 1 iterate over the set of spin
Section 2.1, we introduce the Variational Quantum MOs that can be either composed by Hartee-Fock
Eigensolver (VQE) in the context of Quantum Com- orbitals, or other types of orbitals such as Localized
puting formalism. The main physical quantity used Orbitals (e.g. Boys [31], Pipek-Meyez [32],Edmiston-
to build our ansätze Quantum mutual-information Ruedenberg [33], etc.) or Natural Orbitals [34].
measure is defined in Section 2.2. Then, the formal- By exploiting the Jordan-Wigner mapping [35], each
ism from which we defined one of the basis sets used fermionic operator can be mapped from the fermionic
in this work, the Iterative Natural Orbitals, is de- space to the space of the qubit, thus as a combination
fined in Section 2.3. The extrapolation of the QMI of quantum-gate operations. In particular,
matrices with SparQ is presented in Section 2.4. The i´1
ź
full framework of Molecular-System Multi-QIDA is a:i “ Q:i Zj (2)
shown in Section 3. In particular, the entangling j“0

3
and newly calculated set of parameters, and used to gen-
i´1
ź erate Dthe new circuit. In general, the wavefunction
ai “ Qi Zj , (3)
ψpθ̄q is obtained by applying a unitary transforma-
j“0
tion U pθ̄q to an initial reference state |ψref y. This
where Q:i “ 21 pXi ´ iYi q, Qi “ 12 pXi ` iYi q, and i reference can be either defined as the occupation of
iterates over the set of MOs. They can be associated the spin-MOs according to the HF determinant, or
to creation and annihilation operators in the qubit generated by fixed circuit [37]. The unitary transfor-
space, respectively. As well as the fermionic creation mation U pθ̄q is defined using a PQC.
and annihilation operators act to modify the occu-
pancy of the relative spin-MO, the qubit operators 2.2 Quantum Mutual Information
act to change the state of the qubit related to the
specific spin-MO. The full correspondence is final- The Von-Neumann Quantum Mutual Information
ized with a series of Pauli-Z matrices that allow to (QMI) [38] is a property used to measure the corre-
compute the parity of the state and account for the lation that can be found between two different com-
fermionic anticommutation between a and a: . Us- ponents of a quantum system. Considering a system
ing Equation 2 and Equation2 in Equation1, the new of N elements for which exist a Hilbert space, Hu ,
Hamiltonian defined in the qubit space, Ĥ, can be associated with each element u P t1, . . . , N u, we can
written as define the state of the system as the composition of
these spaces. The resulting total space composition,
L L Nâ
MO
ÿ ÿ denoted with |Ψy, belongs to the composed Hilbert
Ĥ “ ci P̂i “ ci σ̂ji , (4)
j“0
space, Htot , obtained by the tensor product of each
i i
subsystem Hilbert space.
where each σ̂j P tX, Y, Z, Iu is a Pauli matrix acting
N
on the j-th qubit, ci is a coefficient relative to the â
|Ψy P Htot “ Hu . (7)
i-th Pauli string P̂i , NM O is the number of qubit or u“1
4
number of spin-MOs, and L, proportional to NM O,
is the number of Pauli string composing the qubit To each quantum state |Ψy, we can define the asso-
Hamiltonian. ciated density matrix (or density operator)
Now, the Variational Quantum Eigen-
ρ “ |ΨyxΨ| . (8)
solver(VQE) [1] can be used to solve the electronic
structure problem. VQE is an hybrid quantum- Starting from a density matrix of a quantum state,
classical algorithm that can be used to minimize the we can retrieve information about only a subset of
expectation value the elements, k Ă B “ t1, ¨ ¨ ¨ , N u, defining the rela-
tive reduced density matrix (RDM). The RMD of the
Epθ̄q “ xψpθ̄q|Ĥ|ψpθ̄qy, (5)
subset k is computed from the full density operator,
which relies on the Rayleigh-Ritz [36] variational ρ, by tracing out the indices that are not the selected
principle ones. We define k̄ “ B ´ k as the set of elements that
xψpθ̄q|Ĥ|ψpθ̄qy ě E0 (6) have to be traced out. Thus, the RDM of the subset
k, ρk , is formally defined as
in order to estimate the true groundstate energy E0 .
The preparation of the parametrized trial state ψpθ̄q ρk “ trpρqk̄ . (9)
is done on the quantum computer, as well as the mea-
surement of each Pauli string P̂i , while a classical In order to compute the QMI matrix for the quan-
computer is used to compose the complete value of tum system under study, for each component, u, and
the cost function Epθ̄q and optimize the set of pa- each pair of components, pu, vq, we need to compute
rameters θ̄. The trial state is then updated with the the one and two element RDM respectively. Then,

4
once all the reduced density matrices are obtained,
we can compute the Von-Neumann entropy for each
of them. The Von-Neumann entropy, S, is defined as 0
5
Spρq “ ´ Trpρ log ρq. (10) 10
15
Following Equation (10), we can define:
20

Qubit Index
Su “ Spρu q “ ´ Trpρu log ρu q (11) 25
30
and 35
Su,v “ Spρu,v q “ ´ Trpρu,v log ρu,v q, (12) 40
where @u, v P t1, . . . , N u. Finally, from the above 45
equations, we can define the Quantum Mutual Infor- 50
mation Map (or Matrix), I, which is formally defined 55
as
0 5 10 15 20 25 30 35 40 45 50 55
Iu,v “ pSu ` Sv ´ Su,v qp1 ´ δu,v q Qubit Index
(13)
“ pSpρu q ` Spρv q ´ Spρu,v qqp1 ´ δu,v q
Figure 1: Example Quantum Mutual Information map
of an C6 H6 approximated wavefunction with CC-PVDZ
where u, v P t1, . . . , N u. The QMI can be visual-
basis and CAS(30,30). We used Jordan-Wigner mapping
ized with Quantum Mutual Information Maps which
(spin-up orbitals from qubit indexes 0 to 29 and spin-
are symmetric matrices with zero-valued entries on down orbitals from qubit indexes 30 to 59).
the diagonal by definition, shown in Figure 1. QMI
found applications as a quantity to describe the elec-
tronic structure of a molecular system [39]. This mea-
is diagonal. From the diagonal terms of the RMD,
sure does not solely quantify the entanglement a sys-
suppose ρu,u , we can obtain information about the
tem includes, but it takes in account both quantum
number of electrons in the u-th orbital and this value
and classical correlation. a value that describes only
is denoted as Natural Orbital Occupation Number
quantum correlation. We can consider the values of
(NOON) of orbital u.
the I matrix as a measure of the total correlation be-
tween elements of the system, e.g. qubits, orbitals, As claimed in [34], in the NOs basis, the result-
or spin sites as used in this work. ing CI expansion of the state under study is com-
posed by the minimal number of Slater Determinants.
The reduction of the population of SDs of a reference
2.3 Natural Orbitals state is reflected in an increased sparsity of the Quan-
One of the possible way in which the complexity of tum Mutual Information map. The simplification of
the circuit is reduced is exploiting Natural Orbitals the variational problem by means of Natural orbitals
(NOs). In particular, we use them as one-electron ba- in quantum computing has been studied in different
sis set functions for both VQE simulations and QMI works [23, 30, 40].
calculations. For the recursive nature of the basis i.e. the NOs
Given a wavefunction Ψ, the associated NOs are depend on the wavefunction, which is itself defined
defined as the set of molecular orbitals (MOs) for by the NOs, constructing a CI wavefunction in this
which the one-body reduced-density matrix (RDM) basis is tricky. To address this, an iterative procedure
@ ˇ ˇ D known as Iterative Natural Orbitals (INO) [41] can
ρu,v “ Ψˇa:u av ˇΨ (14) be used, which aims to converge to MO where the

5
wavefunction results in a diagonal one-body RDM. 3 Multi-QIDA iterative
Achieving self-consistency with this iterative method
is in principle costly, albeit the convergence rate is
method for shallow ansatz
usually fast, so the process can be halted once a con- Multi-QIDA approach refines the initial QIDA [23]
vergence criterion is achieved. method by constructing ansätze through a multi-
threshold procedure based on approximate QMI ma-
trices, which enables more accurate and efficient
variational quantum simulations. QIDA simplifies
2.4 Retrieving QMI with SparQ
quantum circuit construction by using QMI to pre-
determine where correlations between qubits are ex-
SparQ, or Sparse Quantum State Analysis, is an
pected. This helps in creating circuits that are both
innovative tool designed to efficiently compute key
efficient and effective for specific quantum problems,
quantum information theory metrics for wavefunc-
particularly in reducing the computational resources
tions that are sparse in their definition space [30].
needed for VQE. Multi-QIDA applies the QIDA pro-
SparQ is particularly focused on wavefunctions de-
cedure multiple times to also incorporate mid-low
rived from Post-Hartree-Fock methods and employs
correlations. The application of the Multi-QIDA [29]
the Jordan-Wigner transformation to map fermionic
method demonstrates its utility in quantum simu-
wavefunctions into the qubit space. For excitation-
lations for strongly correlated lattice spin models
based wavefunctions, this is done by applying excita-
like the Heisenberg model, as reported in a previous
tions to the Hatree-Fock SD by the following equa-
work [29]. The procedure is composed of the follow-
tion:
ing steps:
ź
â:i â:j âk âl |HF y “ â:i â:j âk âl â:s |∅y, (15)
sPHF
QMI matrix (SparQ)
ȁ𝝍ۧ𝒓𝒆𝒇
where the indexes i, j P V irtual, k, l P Occupied,
(CISD, MP2, etc…)
and s is the set of occupied orbitals that defines the
Hartree-Fock SD. The fermionic to qubit mapping
Finesse-Ratio
is then directly applied to the excitation operators.
Layer-Building Procedure
This approach leverages the inherent sparsity of these
wavefunctions to perform efficient quantum informa-
tion analysis. This makes it possible to handle larger
and more complex chemical systems than traditional Selection
methods such as the Density Matrix Renormaliza-
tion Group (DMRG) even if sacrificing at times the
quality of the wavefunction compared to the latter
method.
Layer-Independent
The sparsity of the wavefunction strictly depends
ȁ𝝍ۧ𝒐𝒑𝒕 Optimization
on the Post-HF method in use, however, it is mostly Full Layers
related to these methods exploiting only a relatively Relaxation
low number of excitations in the overall Fock space.
Given a computational cost linear in the number Figure 2: Multi-QIDA approach schematic workflow.
of states, SparQ is able to handle any wavefunction
given by the current methods, proving an invaluable
tool for the aim of the present work. 1. QMI calculations: The first step is to build the

6
 approximate QMI matrix exploiting the SparQ accurately. Benchmarks showed improvement in
method, as defined in [30]. (Section 2.4) both energetic terms and in accuracy compared
to other ladder ansatz.
2. Layer-Building Procedure: Without discerning
between classical and quantum correlation, we • Mitigation of Barren Plateaus: Following
spilt the qubit-pairs based on a selected set of the idea that a multi-layer construction of the
QMI values, namely finesse-ratios. For each ansatz may create a funnel in the parameter
range of QMI we obtain a QIDA-layer by per- space that can guide the minimization process,
forming a selection of only some relevant pairs. we may argue that Multi-QIDA’s iterative struc-
(Sections 3.2 and 3.3) ture may also increase the probability of avoid-
ing barren plateau. Barren plateaus are a com-
3. Layer-wise incremental VQE : Each QIDA-layer
mon issue in variational quantum algorithms
is independently optimized, and then the full cir-
where the optimization landscape becomes flat,
cuit goes through a relaxation procedure. (Sec-
complicating the parameter optimization pro-
tion 3.4)
cess. By breaking down the variational land-
The detailed workflow is shown in Figure 2. scape into manageable steps and refining param-
Key aspects of Multi-QIDA’s procedure are: eters in stages, Multi-QIDA achieves better re-
sults compared to HEA ladder ansatz in which
• Layered Ansatz Construction: Multi-QIDA the full parameters space is defined since the be-
constructs variational layers step-by-step, where ginning.
each layer is informed by the QMI matrix, select-
ing qubit-pairs based on their QMI values. This
incremental addition of layers allows for the cap- 3.1 SO(4) as correlators
ture of crucial correlations that single-threshold In the present work we focused only on the usage
approaches might miss. of more complex and expressible gates, instead of
• Efficient Resource Management: The algo- CNOTs, for the construction of the Multi-QIDA cir-
rithm is designed to reduce the computational cuit. The fully parametrized SO4 gate, from the
overhead typically associated with ladder-style O(4), group has been used in each QIDA-layer as
heuristic ansatzes, especially standard Hardware an entangling gate. They offer a tunable correla-
Efficient Ansätze. By selectively entangling tion that can be chosen by the optimizer, as opposed
qubits with strong correlations, Multi-QIDA re- to CNOTs-only-based ansätze in which the correla-
duced the number of required entangling gates, tion is added similarly to a on/off switch. Inside
effectively constructing shallower circuits with- the possible transformation, a SO4 gate can also be
out sacrificing accuracy. This approach could be parametrized to perform an identity, the fermionic-
particularly valuable for current quantum hard- swap gate, which allows to impose the fermionic anti-
ware, where circuit depth and gate count directly symmetrization to the ansatz, as well as Given ro-
impact performance due to noise. tations, which have been used in multiple Quan-
tum Computing based Quantum chemistry applica-
• Improved Convergence and Accuracy: The tion [42]. The identity plays an important role, since
iterative approach embedded in Multi-QIDA al- it can be the starting gate after the addition of a new
lows for faster convergence to the ground-state layer. The choice of SO4 correlator is also justified
energy with fewer optimization runs. It con- as being the most general real-valued 4x4 matrices,
sistently outperforms traditional ladder ansatz performing only real transformation on the wavefunc-
methods by maintaining high precision with re- tion, in line with the fact that the electronic Hamil-
duced mean energy deviation, demonstrating its tonian is composed only of real terms.
effectiveness in calculating ground-state energies A generic gate U P SO(4) is composed by two

7
 is minimized. Thus, if T “ te1ř , e2 , ¨ ¨ ¨ , e|V |´1 u,
then the total weight wpT q “ ePT wpeq is the
lowest possible value among all the available
spanning trees that can be built from the graph
G.
Figure 3: On the left, the circuit implementing a gen-
eral SO4 gate. The gates A and B are general SU2 In the same way, a Maximum Spanning Tree (MST)
parametrized gates. On the right, the symbol we adoptedcan be defined following the previous properties by
in this work for such gate. changing only the total weight values, which in this
case, it has to be maximized. Thus, an MST is a
subset T Ď E that spans all the vertices V , it forms
generic one-qubit rotations A, B P SU(2), four S
a tree, and the total weight wpT q is maximized.
gates and two R gates. It is also known that every
matrix M P SU(2) can be written as a composition In this work, three different weight function has
of Rz pαqRy pθqRz pβq for some α, β and θ, while the R been used:
gate is defined as Ry pπ{2q and the S gate is obtained Maximum Correlation Spanning Tree (MCST): for
with Rz pπ{2q. The U gate is then parametrized using each edge e the associated weight we is defined di-
the two sets of three parameters of gates A and B. rectly as the QMI value of the two vertices v, u that
they are connected by the edge

3.2 Selection criteria we “ wppu, vqq “ Iv,u .

As stated in [29], one of the main components of
Thus, with this type of weight function, we are going
the construction of PQC following the Multi-QIDA
to select the MST that collects the highest amount
method, is the request for a selection criterion for
of correlation.
the reduction of the number of entangling pairs that
fall into each layer. The objective of this procedure is
to exploit cross-entanglement built among the Multi- 4 3 2 4 3 2
QIDA layer in order to reduce the number of corre- 5 1 5 1
lators.
In this work, we have exploited Minimum Spanning 6 0 6 0
Trees (mST) to select which correlators are going to 7 11 7 11
be used in each Multi-QIDA layer. We start by defin- 8 9 10 8 9 10
ing a weighted graph, G “ pV, E, wq, where V is the
set of the vertices, E P V ˆ V is the set of the edges, (a) (b)
and w : E Ñ R is a function that maps each edge
Figure 4: Pictorial representation of the action of the
ei P E to a real valued weight wei . We can define a
correlators reduction. The width of each edge corresponds
MST, T Ď E, which is a subset of the edges of the
to the value of QMI between the two qubits, Iuv . In the
graph G that satisfies the following properties: two figures: (a) Original layer without any reduction ap-
plied. (b) Maximum correlation reduction applied leading
• Vertices Span: The subset of the edges T covers
to a QIDA-layer composed of the remaining edges.
all the vertices V . Formally, for each pair of
vertices u, v P V , there exists a path in T that
allows to reach v starting from u, and vice versa.

• Tree structure: T is an acyclic connected graph. Distance Reduction Spanning Tree (DRST): For
each edge e, the relative weight is defined as
• Total Weight Minimizing: Given the weight
function w, the sum of the weight in the subset T we “ wppu, vqq “ dpu, vq,

8
where dpu, vq is a topology-based distance function. 3.3 Layer-building construction
The distance term is defined as the topological dis-
tance between qubits u and v, which can be seen as As explained in the previous section, our goal is to
the number of edges that are included in the short- create a shallow-depth circuit for state preparation,
est path to connect these two qubits. From another improving the results obtained by the original QIDA
point of view, this distance dpu, vq can be defined as method, in order to include a wider spectrum of dif-
the minimum number of 2-qubits SWAP gates needed ferent qubit-pair correlation. In this section, we will
to make qubits u and j next-neighbour. For this spe- briefly define the pseudo-code to define each layer,
cific case, the topology is linear, thus the distance which is a slight modification of the one presented
function is defined as |u ´ v|. This weight function in [29].
implemented on linear topology corresponds to the The selection is carried out on the list of descending
empirical reduction used in [23], i.e. the objective QMI-value order qubit-pairs obtained from the QMI
is to consider qubit-pairs as close as possible to the matrix Iu,v . Using a threshold µ, called finesse-ratio,
diagonal. Thus, it is required to build a Minimum we can decide the size of the chunks of qubit-pairs
Spanning Tree. that a Multi-QIDA layer has to contain. The finesse-
ratio is not computed automatically with a fixed step,
it is instead empirically determined based on the dis-
tribution of the qubit-pairings. By observing the dis-
tribution of the QMI spots, it can be noticed that
4 3 2 4 3 2 it decays rapidly, concentrating on a minor group of
5 1 5 1 highly/mid correlated spots (as Shown in Figure 6),
6 0 6 0 and thus, as the lower the interval of correlation is
chosen, the higher the number of pairings included in
7 11 7 11 the chunks.
8 9 10 8 9 10
1.0 1.0
(a) (b)
0.8 0.8
Figure 5: Pictorial representation of the action of the 0.6
0.6
QMI

correlators reduction. The width of each edge corre-

sponds to the topological distance between the two qubits, 0.4 0.4
dpu, vq. In the two figures: (a) Original layer without 0.2 0.2
any reduction applied. (b) Empirical reduction leads to
a QIDA-layer composed only of the identified edges. 0.0 0 20 40 60 80 100 0.0
QubitPairs
Figure 6: Example of the decreasing values of QMI for
NH3 INOs molecular system.
In Figures 4 and 5, two examples of correlation re-
duction are shown. In particular, they represent how In section 5.1, the finesse-ratios used for each sys-
a candidate set of entangling gates, image (a) in both tem are shown, for explanation purposes only, we are
figures, gets reduced by the application of the selec- going to call µ̄ the list of finesse-ratios used to create
tion criteria. The weights associated with each edge a Multi-QIDA ansatz.
are defined randomly to be general. Exploiting these For each pair of consecutive finesse ratios, we define
two selection criteria, in the next section, we illus- the range, l, in which the qubit-pairs are used to im-
trate in pseudo-code the algorithm used to construct pose an edge on a graph Gl . This range corresponds
each Multi-QIDA layer. to a Multi-QIDA layer. Selected all the pairings that

9
Algorithm 1 Schematic outline of the Multi-QIDA and in CNOT cost.
Layers-builder All QIDA-layers and the final ladder layer are then
Input: Iij , µ̄, Nqubits , wp¨, ¨q assembled into a complete set of layers, denoted as
Output: List of entangling map L L . This collection L is subsequently passed to the
1: L Ð empty list layer-wise iterative VQE algorithm, which uses it as a
2: m Ð 0 blueprint to construct the adaptive quantum circuit.
3: for m P r0, ¨ ¨ ¨ , lenpµ̄q ´ 1s do
4: G Ð pV “ t1, ¨ ¨ ¨ , N u, E “ Hq 3.4 Incremental VQE optimization
5: for qu , qv P t@Iu,v : µ̄rms ą Iu,v ě µ̄rm ` 1su
do To optimize the full Multi-QIDA circuit, the opti-
6: G.add edgepqu , qv , wpu, vqq mization is defined as an incremental routine with
the aim to not starting with a complete and complex
7: T Ð ComputeM ST pGq
circuit, but instead adding and optimizing only one
8: L.appendpT.get edgespqq
QIDA-layer at a time.
9: return L The selection of a suitable optimization routine is
necessary when dealing with different circuit layouts
in which the composition of the ansatz may vary from
fall in the selected QMI value range, for each u, v in layer to layer. As the name of this section suggests,
the chunk, we define a weighted edge that connects this will be done iteratively along many steps, each of
vertex u to vertex v, with weight wpu, vq. The weight which will include two main phases, an optimization
function wp¨, ¨q depends on the selection criteria ac- of the single layer Ll at the l-th step, and a global
cording to the one defined in the previous section optimization of all the previous layers t0 . . . , l ´ 1u.
(Section 3.2). Once all the edges are inserted in the We remind the reader that each layer Ll is composed
graph, we create the MST (or mST), Tl , concerning of SO4 gates as correlators; however, the procedure
the cost function used to reduce the number of cor- explained here is independent of this variable.
relators. From Tl , we can now retrieve the collection Different from our previous work [29], for these
of the edges EpTl q Ď EpGl q that will compose the molecular systems, the initialization to the identity
entangler map for the l-th layer. of the additional layers turned out not being the
At each step, the graph Gl is reset, ensuring that right choice, due to the presence of local minima, pre-
only the correlators contained within a given chunk venting the optimization from proceeding. To over-
are considered. This guarantees that the minimum come these limitations, we decided to initialize the
spanning tree (MST) is constructed solely from the new layer using an operator that is close to identity
correlators relevant to that chunk, thereby preventing but not identity. This goal is achieved by adding ad-
any edges from previous layers from being reused or ditional layers which has a random offset from the
exploited. identity. The set of initial parameters for the ad-
For each tree Tl , the list of edges forming the MST ditional layer, θ̄li , is created by randomly sampling
(denoted as mST) is translated into a QIDA-layer. values from a uniform distribution with mean 0 and
Once all such layers are generated, a final ”ladder standard deviation 0.1, i.e. θ̄li PR Up0, 0.1q. The
layer” is appended. This layer arranges the remaining value 0.1 has been empirically estimated by selecting
correlators in a top-down topology, allowing separate the lowest value that allowed escaping the local min-
correlation groups to be interconnected. Function- ima of the previous QIDA-layer, while recovering in
ally, the ladder layer acts as a final selection stage for a few VQE iterations the previous energy after the
qubit pairs that fall outside the specified finesse-ratio addition of the successive layer. Now, we can briefly
range. By applying MST-based selection to these define the two steps required to optimize the circuit
leftover pairs, the resulting topology of this layer ap- after the addition of the l-th QIDA-layer: We first
proximates that of a ladder both in term of structure perform an independent optimization of the unitary

10
transformation Ul pθ̄li q, that add the new layer to the Algorithm 2 Iterative (Re)-Optimization routine
previous solution. The action of the unitary is defined Input: N
qubits ą 0, List of entangling map L.
as |Ψl y “ Ul pθ̄li q |Ψl´1 y, where |Ψl´1 y is the previ- Output: Optimal parameters θ̄˚ , Converged en-
tot
ous solution state. The initial parameters are sample ergy Etot
as previously indicated, and after the optimization, 1: QCempty ÐQuantumCircuit(Nqubits )
we obtain the set of optimal parameters for the l-th 2: for l P L do
QIDA-layer, denoted with θ̄li˚ . We then perform a re- 3: add l ÐTrue
laxation of the full circuit. The relaxation is defined 4: while add l do
by 5: append(QCempty , lq
j“0 j“l
ź :
r
ź
r 6: if l “ 0 then
xΨ0 |r Uj spθ̄l q|H|r Uj spθ̄l q|Ψ0 y, (16)
j“l j“0
7: θ̄0 PR r0, 2πq
˚
8: E0 , θ̄tot ÐVQEpQCempty , θ̄0 q
where |Ψ0 y is the initial or reference state, H is the 9: else
Hamiltonian
śj“l in the qubit space, the product of uni- 10: θ̄l PR Up0, 0.1q
taries r j“0 Ui s is the empty circuit up to the l-th 11: El , θ̄l˚ ÐVQEpQCprev , θ̄l q
r r˚ i˚
layer, and θ̄l “ θ̄l´1 ` θ̄l is the concatenation of 12: ˚
θ̄tot Ð θ̄prev ` θ̄l˚
optimal parameters of the relaxation procedure from 13: ˚
Etot , θ̄tot ÐVQEpQCempty , θ̄tot q
the previous layer, l-1, and the optimal parameters ˚ ˚
14: θ̄prev Ð θ̄tot
of the independent optimization of layer l. After the ˚
relaxation procedure, the set of optimal parameters 15: QCprev ÐassignpQCempty , θ̄prev q
˚
up to the layer l is defined and denoted with θ̄r˚ . return E , θ̄
tot tot
l
To briefly resume a general procedure that is exe-
cuted at each step l, exploiting non-fixed parameters
quantum circuit, i.e. QCempty : optimization method for other iterative ansätze, as it
is a similar procedure to the one used by the Adapt-
1. Append the l-th layer to the QCempty ; VQE algorithm [9], as well as in Layer-VQE [43].
2. Assign the optimal parameters up to the previ-
˚
ous iteration θ̄0,1,...,l´1 “ θ̄prev ;
4 Computational details
3. Initialize the parameters of the l-th layer to an
offset of the identity; 4.1 Molecular Systems
4. Find the optimal parameters of the l-th layer
To test the Multi-QIDA approach on molecular sys-
alone, θ̄l˚ ;
tems, we considered five different molecules that,
5. Compose the total set of parameters up the the once codified on the quantum computer, span the
˚
l-th layer as θ̄tot “ θ̄prev ` θ̄l˚ ; range between 8 to 14 qubits. The systems chosen
are H2 O, NH3 , and BeH2 in terms of full-size system,
6. Use θ̄tot as starting parameters for a final VQE thus Full-CI level, while H2 O and N2 with bigger ba-
in which the variational wavefunction is defined sis have been studied at CASCI level. In Table 1, we
by QCempty ; have summarized all the information related to the
system under study. Only the frozen core approx-
7. Once converged, the optimized set of the com-
˚ imation at the Hartree-Fock level has been used for
bined circuit, i.e. θ̄tot is obtained.
the first group of three molecules, by freezing the first
We notice that the optimization procedure shown core orbital for each system. All the systems in the
in Algorithm 2 is not limited to the use in combina- first group have been analyzed with STO-3G basis
tion with Multi-QIDA, but could also be used as an set. One HF is computed, we have applied the pro-

11
 Mol. Coordinates(Å) Basis Qubits correctly appear lower in energy than the πg˚ , leading
H 0.757 0.586 0.0 INOs to an underestimated HOMO-LUMO gap. With cc-
H2 O H -0.757 0.586 0.0 RCISD 12 pVTZ, the active space selection better captures the
O 0.0 0.0 1.595 (STO-3G)
Be 0.0 0.0 1.334 INOs
electron correlation essential for describing the triple
BeH2 H 0.0 0.0 0.0 RCISD 12 bond in N2 . In minimal sets like STO-3G, improper
H 0.0 0.0 2.668 (STO-3G) orbital energy ordering may lead to inaccuracies in
N 0.0 0.0 0.1211
INOs multiconfigurational calculations [44].
H 0.0 0.9306 -0.2826 Starting from the set of Mact , we can define the sec-
NH3 RCISD 14
H 0.8059 -0.4653 -0.2826
H -0.8059 -0.4653 -0.2826
(STO-3G) ond quantization Hamiltonian Ĥ (Equation 1), ap-
plying Jordan-Wigner mapping to obtain the elec-
H 0.847 0.0 0.0
H2 O tronic molecular Hamiltonian in the qubit space, Ĥ,
H -0.298 0.0 0.793 6-31G 8
CAS(4,4) as defined in Equation 4. The number of qubit re-
O 0.0 0.0 0.0
N2 N 0.0 0.0 -0.5488
cc-pVTZ 12
lated for each system will be 2 ˆ Mact , split between
CAS(6,6) N 0.0 0.0 0.5488 |Mact | spin-α qubits and |Mact | spin-β qubits, with
ordering |α . . . αβ . . . βy.
Table 1: Molecular systems under analysis. The num-
ber of qubits is computed as 2*Mact , where Mact are the
active orbitals. For the first three system, the basis-set in 4.2 Heuristic Ansätze Comparison
brackets is the initial one on which INOs are constructed.
We decided to compare our approach with the most
general variational wavefunction as the Hardware-
Efficient Heuristic ansatz. In particular, the way
cedure to obtain INOs for each system, starting from in which the correlators are placed for this type of
RCISD calculations. The INOs obtained are used as ansätze is in ladder fashion: a sequence of rotation
set of MOs that will define the true active space of gates is followed by a series of CNOTs placed in a
the system, i.e. Mact . top-ordering connecting adjacent qubits, this config-
For the second group of molecules instead, a more uration repeated d times, and completed by a final
fine selection is performed. For H2 O, the CAS(4,4) series of parametrized rotation gates. The depth of
active space includes 4 electrons in 4 orbitals: two σ the ladder has been chosen to be of the order of mag-
bonding (O-H) and two σ ˚ antibonding orbitals. Us- nitude of Multi-QIDA CNOT counts. The number of
ing the 6-31G basis set provides a moderate descrip- CNOTs for ladder fashion circuits is defined as pN -
tion of the molecular orbitals, accurately represent- 1q ˚ d, where N is the number of spin-orbitals, and
ing the bonding (HOMO) and antibonding (LUMO) d is the depth. HEA are denoted with pLqCX 5 , while
levels. For this system, the selected orbitals will com- Multi-QIDA using the label QIDAsel , where sel can
pose the set Mact with size 4. For N2 , the CAS(6,6) be one of the two selection criteria defined in Sec-
active space involves 6 electrons distributed across tion 3.2. In particular, the complete CNOTs count is
6 orbitals: bonding σg p2pz q, πu p2px q, πu p2py q, and shown in Table 2.
antibonding σu˚ p2pz q, πg˚ p2px q, πg˚ p2py q. For the N2 ,
the set of Mact will be composed of 6 elements. Us-
4.3 Metrics and Measures
ing the cc-pVTZ basis set, the orbital energies are
more accurate due to better flexibility and polariza- The main metric used to compare different ansatz
tion functions, resulting in a realistic HOMO-LUMO configurations is the number of CNOTS, #CNOT.
gap. In this case, the HOMO is degenerate, includ- We decided to not employ the measurement of the
ing both the πu p2px q and πu p2py q orbitals, as well depth of the circuit, due the fact that for the Multi-
as the LUMO, composed by πu˚ p2px q and πu˚ p2py q. QIDA method, the full circuit is composed by entan-
In contrast, the STO-3G basis set often predicts a gler maps that differs from layer to layer. Thus, a
different ordering, where the σu˚ p2pz q orbital can in- different number and disposition of the correlators,

12
 H2 O N2 MCED is formally defined as
#CNOTs BeH2 H2 O NH3
CAS(4,4) CAS(6,6)
ř#V QEs
pLqCX
d 66 55 65 35 66 |ϵi ´ ϵbest |
QIDAmax 70 58 66 36 68 M CED “ i , (19)
#V QEs
QIDAemp 70 58 66 36 68
where ϵi is the correlation energy for a specific VQE
Table 2: Number of CNOTs used by each ansatz config- run, ϵbest is the correlation energy of the best per-
uration. pLqCX
d denotes HEA, while QIDAsel to different forming simulation, and #V QEs is the total number
Multi-QIDA ansätze. The number of CNOTs in Multi- of simulations for a given ansatz configuration. All
QIDA configuration is equal due to the selection based on
the results related to average and best-performing
spanning trees.
simulations for each ansatz configuration are col-
lected in Table 3 for INOs systems, and in Table 4
leads to an inhomogeneous metric. To measure the for Active-Space systems.
performance of the variational calculation, we used We also computed four more quantities for each
used percentage correlation energy, ϵ, defined as wavefunction computed by both Multi-QIDA and
HEA circuits. We are interested in :
EV QE i ´ EHF
ϵi “ 100 ¨ • Fidelity with the true ground-state:
EF CI ´ EHF
(17)
xψpθ̄qi |Ĥ|ψpθ̄qi y ´ EHF F “ xΨGS |ψpθ̄qi y, (20)
“ 100 ¨
EF CI ´ EHF
where i is the i-th VQE simulation.
where EV QEi is the converged energy of the i-th sim-
ulation, EHF is the Hartree-Fock SCF energy, while • Projection along z-axis of the spin:
EF CI is the exact solution, obtained by performing 1
a diagonalization on the qubit Hamiltonian defined Ŝz “ pN̂α ´ N̂β q
2
on the Mact orbitals, and selecting the lowest eigen- Mact Mact (21)
1 ÿ ÿ
value. The EHF is instead directly obtained by the “ p a:i,α ai,α ´ a:i,β ai,β q.
RHF solver of PySCF [45–47] package. 2 i i
For the system in which a specific active space is se-
lected i.e. H2 O-CAS(4,4) and N2 -CAS(6,6), the ref- • Spin squared which can be defined as
erence exact energy correspond to the CASCI energy,
diag Ŝ 2 “ Ŝ´ Ŝ` ` Ŝz pŜz ` 1q, (22)
ECASCI “ Ecore ` Eact , where Ecore is the energy
diag
contribution of inactive occupied orbitals, and Eact where
is the lowest eigenvalue of the active Hamiltonian. M act M act
ÿ ÿ
Equation 17 can be redefined considering an active Ŝ´ = a:i,β ai,α and Ŝ` = a:i,α ai,β ,
Hamiltonian as i i
EV QEi ´ EHF
ϵact “ 100 ¨ . (18) • Number of particles in this case, electrons,
ECASCI ´ EHF obtained from
We have then computed the Mean Correlation En- ÿ M
ÿact
ergy Deviation (MCED) which quantifies the average N̂e “ a:i,σ ai,σ . (23)
deviation of the correlation energy from the best per- σPtα,βu i
forming simulation ϵbest . It is obtained by summing
up the difference between the correlation energies of All the four additional properties measured, the val-
each VQE simulation and the correlation energy of ues of best-performing results, are collected in Table
the best performing simulation, then, the sum is nor- 5 for INOs systems, and in Table 6 for Active-Space
malized by the total number of simulations. The systems.

13
4.4 Simulation details for SparQ are a cutoff of 10´12 for the Slater Determi-
nant (SD) coefficients, and a maximum of 105 SDs to
The part of classical Quantum Chemistry included build the approximated wavefunction. Then, for each
in this work is performed with the PySCF Python system, by observing the distribution of the mutual-
package. The computation of the INO is made us- information pairs, the selection of the finesse-ratios is
ing Restricted-CISD (RCISD). We used the QuAQ performed. Starting from the QMI matrices, shown
(Quantum@L’Aquila) [48] code base for most of in Figure 7, the different finesse-ratios used are:
the preprocessing part, as well as for the defini-
tion of the Iterative VQE procedure and the Multi- • H2 O INOs: [0.5, 0.3, 0.1]
QIDA layer builder. For the computation of the
QMI map from the reference wavefunction, we used • BeH2 INOs: [0.7, 0.4, 0.35, 0.3, 0.2]
SparQ [30] algorithm, which is actually contained
in QuAQ codebase. For the creation of quantum • NH3 INOs: [0.75, 0.5, 0.25, 0.2]
circuit and anything directly related with them, we
used the Qiskit Python library [49]. We tested our • H2 O 6-31G CAS(4,4): [0.5, 0.20, 0.15]
approach on different molecules, for each molecule
a series of 50 simulations were carried out. We • N2 cc-pVTZ CAS(6,6): [0.80, 0.6, 0.4, 0.2]
employed noiseless statevector simulation. As gen-
eral settings for any VQE, we decided to use the The choice of the finesse-ratio is done accordingly
Broyden–Fletcher–Goldfarb–Shanno (BFGS) [50] al- to the criteria defined in Section 3.2, in particular,
gorithm with a convergence threshold set to 10´6 , we avoid the creation of a highly populated candi-
which corresponds to the tolerance on the gradient date set, we cover all the qubits with at least one
of the parameters. QIDA-layer, and we stop after reaching a layer be-
low 0.2 of QMI value. The number of QIDA-layers
is different for each system, but generally, we need
5 Results to encode at least 3 layers in order to recover highly
correlated pairs, mid-correlation, and low-lying cor-
Multi-QIDA builds a compact reference circuit start- relations. As in the previous work, we completed the
ing from an approximated Quantum Mutual Informa- series of QIDA-layers with an additional ladder in
tion matrix, computed by quantum chemistry meth- order to join together disjointed groups of qubits.
ods, iteratively adding and optimizing a layer defined
on the desired range of correlation strength. Here,
we collect the results obtained from our Multi-QIDA
5.2 Performance analysis
approach compared to standard Hardware-Efficient In this section, the energetic comparison between
Ansatze (HEA) with ladder topology. HEA and Multi-QIDA, with also the deviation from
the best performing VQE of both percentage corre-
5.1 Preprocessing: QMI matrices lation energy and absolute energy, are shown.
The results presented in Table 3 and Table 4
Exploiting the SparQ algorithm, we computed the demonstrate that the proposed Multi-QIDA ap-
QMI matrices for all the systems listed in Table 1. proaches, QIDAmax (reduction of each QIDA-layer
For all the systems, an RCISD wavefunction has been using a MST that maximizes the total QMI value)
used as a reference to build the QMI map. In par- and QIDAemp (reduction of each QIDA-layer using
ticular, starting from the same set of orbitals used a mST that minimizes the distance between each
in the circuit, we used the PySCF RCISD solver qubit), consistently outperform the standard ladder
in order to obtain the relevant information used by ansatz pLqdCX in terms of correlation energy and ab-
SparQ to build the QMI matrix. The settings used solute energy across all tested molecular systems.

14
 0 2 4 6 8 10 0 2 4 6 8 10 BeH2
0 1.0 0 1.0
pLqcx QIDAmax QIDAemp
2 0.8 2 0.8 6
ϵavg r%s 21.25p14.55q 79.78p10.22q 80.46p9.54q
4 0.6 4 0.6 Eavg rHas -3.9146 ´3.93490 ´3.93510
6 0.4 6 0.4 ϵbest r%s 49.75 90.9 84.03
8 8
0.2 0.2 Ebest rHas -3.92441 -3.93872 -3.93637
10 10
0.0 0.0 M ϵDbest r%s 28.17 10.98 3.57
M EDbest rHas 0.00980 0.00380 0.00120
(a) (b)
0 2 4 6 8 10 12 0 1 2 3 4 5 6 7 H2 O
1.0 0 1.0
0 pLqcx
5 QIDAmax QIDAemp
2 0.8 1 0.8 ϵavg r%s -111.50p284.70q 89.52p2.31q 89.71p2.31q
4 2
0.6 3 0.6 Eavg rHas -23.45440 ´23.55380 ´23.55390
6
8 0.4 4 0.4 ϵbest r%s 81.56 91.51 91.55
5 Ebest rHas -23.54985 -23.55477 -23.55479
10 0.2 6 0.2
12 7 M ϵDbest r%s 193.07 1.99 1.85
0.0 0.0 M EDbest rHas 0.09550 0.00100 0.00090
(c) (d) NH3
0 2 4 6 8 10 pLqcx
0 1.0 5 QIDAmax QIDAemp
ϵavg r%s -71.13p248.89q 54.12p0.36q 54.10p0.36q
2 0.8 Eavg rHas -20.00260 ´20.08510 ´20.08510
4 0.6 ϵbest r%s 41.22 54.95 54.95
6 0.4 Ebest rHas -20.07658 -20.08561 -20.08561
8 M ϵDbest r%s 112.36 0.83 0.85
0.2
10 M EDbest rHas 0.07400 0.00050 0.00060
0.0
(e) Table 3: BeH2 ,H2 O, and NH3 INOs system results.

Figure 7: QMI matrices obtained by the tested systems:

(a) H2 O INOs/12 qubits. (b) BeH2 INOs/12 qubits.
(c) NH3 INOs/14 qubits. (d) H2 O CAS(4,4). (d) N2 correlation energy (´111.50%) from pLq5CX , in-
CAS(6,6). All obtained with SparQ at R-CISD level. dicating that the ladder ansatz optimization is
failing to converge, leading to a state with energy
higher than HF. In this case, the deviation is two
orders of magnitude lower for the QIDA circuit.
• INOs systems: In the simulations of BeH2 , For NH3 , while all methods show similar results,
H2 O , and NH3 , Multi-QIDA circuits consis- QIDAmax and QIDAemp marginally outperform
tently achieve higher average percentage corre- the ladder topology with ϵavg values of ´54.12%
lation energy (ϵavg %). In particular, for BeH2 , and 54.10%, respectively, compared to the neg-
QIDAmax achieves an ϵavg of 79.78%, a signifi- ative value of ´71.13% for pLqdCX . Here, both
cant improvement over the 21.25% obtained us- QIDAmax and QIDAemp obtain close values of
ing the ladder ansatz, with a lower standard de- percentage correlation energy deviation, close to
viation as well (10.22% versus 14.55%). Bet- zero. In terms of best VQE results, BeH2 , QIDA
ter results are obtained by QIDAemp , which in- obtains a clear increase of 40% over the ladders,
creases the ϵavg up to 80.46% Similarly, for H2 O, whereas for the other systems, the increase is
QIDAmax obtains 89.52%, QIDAemp reaches a around 10{12%. In particular, for BeH2 , Multi-
close 89.71%, whose compared to the negative QIDA circuits do not fall far from the average

15
 case, as expected. Compared to the 49.75% ob- tem, QIDAmax achieves an ϵavg of 82.36% while
tained by the ladder, QIDAmax reaches 90.90%, QIDAemp reaches a 80.32%, which are both en-
while QIDAemp gets 84.03%. For the correla- hanced performance with respect to the HEA
tion energy of best performing VQE of H2 O, ladder, which obtains on average 55.42% correla-
we can observe that the HEA reaches a good tion energy. The same behavior can be found for
81.56%, while QIDAmax and QIDAemp get sim- the N2 CAS(6,6), for which the standard ladder
ilar values, around 91%. Finally, for NH3 , the obtains on average 20.85% of correlation energy,
ladder reaches 41.22% correlation energy, while while QIDAmax is able to reach 57.13% and
Multi-QIDA in both settings reaches 54.94%. In QIDAemp , a slightly higher value of 58.48%. In
terms of percentage correlation energy deviation, terms of best-performing VQE results, we have
QIDA shows a lower dispersion w.r.t. the best- that the results of HEA ladders and Multi-QIDA
performing VQE, hitting a dispersion of two or- circuits are close, but in any case, the latter
ders of magnitude lower than ladder ansatz for reaches slightly higher correlation energy.
H2 O and NH3 .
Comparing instead the results obtained in terms of
In general, we can find a consistent number of VQEs percentage correlation energy deviation w.r.t. to the
for the ladder topology, which due to the random best-performing VQE, we can notice that M ulti ´
initial parametrization, are guided in a completely QIDA behaves clearly better for complete systems,
wrong energetic solution, way lower than the HF en- so in our case INOs systems, while it has a higher
ergy. dispersion for Active space systems, which may be
related to the fact that we are not including any kind
H2 O CASp4, 4q
of double excitations directly in the ansätze. Also, we
pLqcx
5 QIDAmax QIDAemp
can observe that there is no clear distinction between
ϵavg r%s 55.42p27.46q 82.36p6.32q 80.32p9.18q
the two type of selection performed on the QIDA-
Eavg rHas ´6.62720 ´6.62800 ´6.62790
layers, a difference that may be appreciated more if
ϵbest r%s 92.17 95.42 97.81
applied in the context of real hardware topology or
Ebest rHas ´6.62828 ´6.62837 ´6.62844
real devices.
M ϵDbest r%s 36.75 13.06 17.49
M EDbest rHas 0.00110 0.00040 0.00050
N2 CASp6, 6q 5.3 Convergence and Precision
pLqcx
6 QIDAmax QIDAemp Here, we briefly analyze the results presented in Fig-
ϵavg r%s 20.85p79.43q 57.13p6.53q 58.48p17.59qure 8 and in the Appendix Figures A1-A4, related to
Eavg rHas ´11.44410 ´11.46860 ´11.46950 the precision of VQE runs, and Figure 9 and in the
ϵbest r%s 79.99 82.17 85.22 Appendix Figures A5-A8, related to the dispersion of
Ebest rHas ´11.48403 ´11.48551 ´11.48757 the optimizations. Each of the plots in the first group
M ϵDbest r%s 59.13 25.04 26.75 represents the energy, E, and the percentage correla-
M EDbest rHas 0.04000 0.01690 0.01810 tion energy, Ecorr , for every system and for all three
ansatz configurations. In these plots, the results for
Table 4: H2 O 6-31G CAS(4,4) and N 22 cc-PVTZ
each ansatz configuration are represented by a vio-
CAS(6,6) system results.
lin plot. The width of each violin is related to the
frequency of the VQE outcomes. They are useful for
• Active Region systems: The second group of sim- assessing the consistency of the algorithm (how clus-
ulations are related to the application of Multi- tered or spread out the results are) and identifying
QIDA in more complex systems, which are con- trends, such as whether the algorithm reliably con-
sidered by dividing orbitals into inactive and ac- verges to a minimum energy or exhibits variability.
tive space regions. For the H2 O CAS(4,4) sys- At the two extremes of each violin are presented the

16
 0 -84.157 QIDA in correctly describing the NH3 ground state
and in particular, it get stuck around 50% of corre-
lation energy, shown in Figure A2. But in general,
20 -84.167 we can see that even in the worst-performing VQE,
for which the population is very small, the energy
is still higher than the average case of HEA and in
40 -84.177 some cases higher that the best-performing ladder

Energy (Ha)
ECORR(%)

VQE. This last case is shown in Figure 8. In Figure

A4, we can see how, for the N2 CAS(6,6) the per-
60 -84.187 formance of the two Multi-QIDA selection criteria is
completely different, and in particular, the maximum
58 58 correlation fails to compact the results towards the
89.45 89.73
80 91.51 91.55 -84.196 best-performing VQE but on the worst-performing,
55 still reaching higher correlation energy than the av-
-108.47
81.56 erage HEA circuits. Another example is for BeH2 , in
which the best-performing for ladders reaches „50%
100 -84.206
correlation energy, while a very limited portion of
(L)CX
5 (M-QIDA)SO4
max (M-QIDA)emp
SO4
VQEs for Multi-QIDA fails to follow the right varia-
Ansatz configurations tional path. Multi-QIDA approach, as already seen
for spin systems, can guide the variational wavefunc-
Figure 8: H2 O INOs system comparison between depth tion in the right spot, in an iterative and adaptive
5 ladder HEA and Multi-QIDA in both max and empirical way, without the requirement of building and opti-
configuration. The three numbers associated with each mizing the full variational space from the beginning.
violin show the number of CNOTs, ϵavg , and ϵbest , This behavior can also be noticed in the second group
starting from the upper one, for each simulation setting.
0
(L)CX
6
All runs
Ecorr(%)

worst and the best performing VQE results, while the 50 Avg
central dot represents the average value. Best
We can notice that standard HEA with ladder- 100
fashion connectivity presents results that are way 0
(M-QIDA)SO4
max
lower than HF and thus the average value is strongly
Ecorr(%)

shifted from the best-performing VQE. In the sys- 50

tems in which this behaviour does not happen, the
average result is anyway distant than the optimal one 100
and this means that it is required to re-run the cir- 0
(M-QIDA)SO4
emp
cuit even more time, compared to Heisenberg Model
Ecorr(%)

Hamiltonian, [29] before obtaining a satisfying result. 50

Ladders quickly encounter and falls into local minima
that are far away from the best result. An example is 100
0 1000 2000 3000 4000 5000
in Figure 8, for which HEA can reach a good 81,54% Iterations
of correlation energy but is heavily penalized on av-
erage because most of the runs falls below HF energy, Figure 9: H2 O INOs system convergence trajectories
while Multi-QIDA is able to maintain, as expected, for each of the 50 VQEs. In particular, starting from the
a low dispersion around the best-performing VQE. upper plot pLq5 , Multi-QIDA with max selection criteria,
We can further identify some difficulties of Multi- and last Multi-QIDA with emp reduction.

17
of plot, Figure 9 and in Appendix Figures A5-A8, in BeH2
which we show the trajectories of the convergence pLqcx
6 QIDAmax QIDAemp
of each VQE for all the systems. For HEA simula- Favg r%s 98.89579 99.60117 99.61037
tions, the upper subplot, it is clear that the popu- Sˆz avg -0.00002 0 0
2
lation of VQE that are actually getting towards the Ŝavg 0.00261 0.00111 0.00062
right optimization path is very low compared to the N̂avg 4.00001 4.00000 3.99999
one that diverges or gets stuck in local minima. For Fbest r%s 99.28972 99.80993 99.65462
Multi-QIDA instead, it is possible to notice that the Sˆz best 0 0 0.00001
2
trajectories, even if they get perturbed at each ad- Ŝbest 0.00016 0.00026 0.00002
ditional layer, tend to be more compact and closer N̂best 4.00000 4.00010 3.99999
to the best-performing and average trajectory. From H2 O
the best-performing VQE, which is the bolder trajec- pLqcx QIDAmax QIDAemp
5
tory, it is also possible to notice the quick recovery Favg r%s 89.08740 99.86803 99.86958
and restart of the optimization after being perturbed, Sˆz avg 0.10343 0 0
allowing the escape from the previous local minima 2
Ŝavg 0.22269 0.00128 0.00115
and without ending in a higher convergence point. N̂avg 7.97648 7.99998 7.99998
As in the previous work, we are aware of the higher Fbest r%s 99.77667 99.88329 99.88125
computational cost required by Multi-QIDA to con- Sˆz best 0 0 0.00001
verge and to end the full optimization procedure. 2
Ŝbest 0.00001 0.00015 0.00042
The average number of iteration is usually two/three
N̂best 8.00000 8.00000 7.99998
times the number of iterations required by the corre-
sponding HEA ansatz, and most of the optimization NH3
procedure is wasted in the relaxation procedure. pLqcx
5 QIDAmax QIDAemp
Favg r%s 92.23381 99.19447 99.19421
Sˆz avg 0.05542 0 0
5.4 Wavefunction Properties 2
Ŝavg 0.13443 0.00003 0.00004
N̂avg 7.98916 8.00000 8.00000
Together with the measurement of the performance,
Fbest r%s 98.98914 99.20485 99.20485
we decided to define also metrics to evaluate the capa-
Sˆz best 0 0 0
bility of the Multi-QIDA ansatz to satisfy symmetry 2
constraints and fidelity w.r.t. the exact ground state. Ŝbest 0.00002 0 0
Given the fact that all the system studied are closed N̂best 7.99998 8.00000 8.00000
shell, the Sˆz and Ŝ 2 are both zero, while the num-
Table 5: Properties of BeH2 , H2 O, and NH3 INOs sys-
ber of particles, N̂e , for each specific INOs system is: tem.
BeH2 =4 , H2 O=6, and NH3 =8 , for CAS system in-
stead: H2 O CASp4, 4q=4 and N2 CASp6, 6q=6. The
properties analysis results are collected in Table 5 for
INOs systems, and in Table6 for Active-Space sys- the spin symmetries, in both cases, Multi-QIDA
tems. has been able to improve the results or in general
to not lower the quality of the result. In partic-
• INOs Systems : On average, Multi-QIDA ob- ular, for Sˆz , Multi-QIDA obtain 0 in the two se-
tained an improvement on each of the property lection criteria, while the HEA gets 0.10343 for
measured. For the Fidelity F with respect to H2 O and 0.05541. In terms of Ŝ 2 instead, the im-
the ground state, both the Multi-QIDA configu- provements have been obtained on all the three
ration recovered a slightly higher value, „ 1.30% systems, in particular, for BeH2 , Multi-QIDA
,for BeH2 , a relevant increment of „ 10.8% for with max selection halved to 0.00111 the HEA
H2 O, and a non-negligible „ 6.8% for NH3 . For value, 0.00261, while Multi-QIDA with emp re-

18
 duce the value lower to 1e-3. For H2 O, the value H2 O CASp4, 4q
for HEA is 0.22268 and Multi-QIDA configura- pLqcx5 QIDAmax QIDAemp
tion reduced it of two orders of magnitude. The Favg r%s 99, 95724 99, 98186 99, 97978
best improvement have been obtained for NH3 Sˆz avg 0 0 0
2
for which the value obtained by HEA, 0.13443, Ŝavg 0, 00025 0, 00035 0, 00030
has been reduced by four orders of magnitude. N̂avg 4, 00000 4, 00000 4, 00000
For the number of particles, N̂ , the values have Fbest r%s 99, 99057 99, 99390 99, 9967
been refined to exact values only for NH3 . In Sˆz best 0 0 0
2
terms of best-performing VQE, as we expected, Ŝbest 0, 00018 0, 00010 0, 00004
also standard HEA is able to recover almost the N̂best 4, 00000 4, 00000 4, 00000
same values of properties as Multi-QIDA. N2 CASp6, 6q
pLqcx
5 QIDAmax QIDAemp
• Active Region systems : For this systems, the
Favg r%s 94, 16438 98, 77033
98, 70980
main improvement of Multi-QIDA with respect
Sˆz avg ´0, 00007 0 0
to HEA can be found mainly for the second sys- 2
Ŝavg 0, 05629 0, 00049
0, 04372
tem, N2 CAS(6,6). On average, Multi-QIDA
N̂avg 6, 04056 6, 00000
6, 00000
obtains an slightly higher fidelity compared to
F best r%s 99, 5714 99, 55640
99, 67421
HEA and for N2 closer Ŝ 2 and N̂ to the exact
Sˆz best ´0, 0002 0 0
value. As before and as we expected, the best- 2
Ŝbest 0, 00192 0, 00766 0
performing VQE of HEA is able to obtain prop-
N̂best 5, 99956 6, 00000
6, 00000
erties values closer to the one measure from a
Multi-QIDA circuit. Table 6: Properties of H2 O 6-31G CAS(4,4) and N2 cc-
PVTZ CAS(6,6) system.

6 Discussion and Conclusions

curacy, while maintaining fidelity to the true ground
The Molecular System-Multi-QIDA method, an ex- state and respecting essential physical symmetries.
tension of the Quantum Information Driven Ansatz Furthermore, by incorporating iterative optimiza-
(QIDA) is designed to leverage quantum mutual tion, Multi-QIDA mitigates the challenges of barren
information (QMI) to construct compact, shallow plateaus, offering scalability and convergence advan-
quantum circuits that are tailored on the main and tages. These features make it a strong candidate for
most relevant correlations present in the system. The use as a starting guess in more complex ansätze like
method generates an initial wavefunction using QMI- ADAPT-VQE or sampling procedures, such as Quan-
derived correlations and incrementally adds layers to tum Selected CI (QSCI).
recover missing correlations, compared to the stand- In conclusion, the Multi-QIDA method demon-
alone QIDA method. Combined with the including of strates significant promise in constructing resource-
SO(4) correlators and spanning-tree based reduction efficient and accurate quantum circuits for molecular
of the qubit-pairs, Multi-QIDA is able to achieve a simulations. However, several open questions remain.
balance between computational efficiency and circuit Can the method maintain its performance and scala-
expressiveness, while guaranteeing reliable results in bility as the size and complexity of molecular systems
terms of good approximated wavefunctions. is increasad. How effectively can Multi-QIDA inte-
This approach has been benchmarked across vari- grate with adaptive approaches like ADAPT-VQE or
ous molecular systems, including H2O, BeH2, NH3, sampling methods to tackle strongly correlated sys-
and active-space models like N2 CAS(6,6) and H2O tems while avoiding optimization bottlenecks. What
CAS(4,4). It consistently outperforms hardware- modifications would be required for Multi-QIDA to
efficient ansatz (HEA) configurations in energy ac- perform robustly on real-world quantum devices sub-

19
ject to noise and decoherence. Can be extended in- 1723. doi: 10.1038/ncomms5213. url: https:
cluding different correlators, such as single/double- //doi.org/10.1038/ncomms5213.
qubit-based excitations or Givens rotations. Lastly, [2] Jarrod R McClean, Jonathan Romero, Ryan
is Multi-QIDA generalizable to problem in which a Babbush, and Alán Aspuru-Guzik. “The the-
correlation matrix can be defined amongst the el- ory of variational hybrid quantum-classical al-
ements of the system. Addressing these questions gorithms”. In: New Journal of Physics 18.2
will further clarify the potential of Multi-QIDA in (2016), p. 023023. issn: 1367-2630. doi: 10 .
advancing quantum chemistry simulations. 1088/1367- 2630/18/2/023023. url: http:
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pirical ansatz, the SO4 gates provides a notable ad- 023023.
vantage not only in term of variational energies but
also in term of preserving wavefunction properties [3] Abhinav Kandala, Antonio Mezzacapo, Kris-
and symmetries, such as total spin, spin projection, tan Temme, Maika Takita, Markus Brink, Jerry
and particle number, thus ensuring better capacities M. Chow, and Jay M. Gambetta. “Hardware-
to describe molecular systems. efficient variational quantum eigensolver for
small molecules and quantum magnets”. In:
Nature 549.7671 (2017), pp. 242–246. issn:
Acknowledgments 1476-4687. doi: 10.1038/nature23879. url:
http://dx.doi.org/10.1038/nature23879.
The authors acknowledge funding from the European
[4] M. Cerezo, Andrew Arrasmith, Ryan Babbush,
Union - Next Generation EU, Mission 4 - Component
Simon C. Benjamin, Suguru Endo, Keisuke Fu-
1 - Investment 4.1 (CUP E11I22000150001). The
jii, Jarrod R. McClean, Kosuke Mitarai, Xiao
authors acknowledge funding from the MoQS pro-
Yuan, Lukasz Cincio, and Patrick J. Coles.
gram, founded by the European Union’s Horizon 2020
“Variational quantum algorithms”. In: Nature
research and innovation under Marie Sklodowska-
Reviews Physics 3.9 (2021), pp. 625–644. issn:
Curie grant agreement number 955479. The authors
2522-5820. doi: 10.1038/s42254-021-00348-
acknowledge funding from Ministero dell’Istruzione
9. url: http : / / dx . doi . org / 10 . 1038 /
dell’Università e della Ricerca (PON R & I 2014-
s42254-021-00348-9.
2020). The authors also acknowledge funding from
National Centre for HPC. Big Data and Quantum [5] Dmitry A. Fedorov, Bo Peng, Niranjan Govind,
Computing - PNRR Project, funded by the Euro- and Yuri Alexeev. VQE Method: A Short Sur-
pean Union - Next Generation EU. vey and Recent Developments. 2021. arXiv:
L.G. acknowledges funding from the Ministero 2103.08505 [quant-ph].
dell’Università e della Ricerca (MUR) under the [6] Kishor Bharti, Alba Cervera-Lierta, Thi Ha
Project PRIN 2022 number 2022W9W423 through Kyaw, Tobias Haug, Sumner Alperin-Lea,
the European Union Next Generation EU. Abhinav Anand, Matthias Degroote, Her-
manni Heimonen, Jakob S. Kottmann, Tim
Menke, Wai-Keong Mok, Sukin Sim, Leong-
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24
A.1 Violin plots and Trajectories Convergence percentage correlation en-
ergy/absolute energy HEA against Multi-QIDA
Convergence percentage correlation energy/absolute configurations for CASCI/Active region molecular
energy HEA against Multi-QIDA configurations for systems.
INOs molecular systems.
0 -86.337
0 -18.932

20 -86.337
20 -18.939

70 70 40 -86.338

Energy (Ha)
40 79.79 80.53 -18.946
Energy (Ha)
90.75 84.03

ECORR(%)
ECORR(%)

66 36 36
21.65 82.45 80.41
49.75 60 95.42 97.81 -86.338
60 -18.953

80 -86.339
80 -18.96
35
55.82
100 92.17 -86.339
100 -18.967
(L)CX
5 (M-QIDA)SO4
max (M-QIDA)emp
SO4
(L)CX (M-QIDA)SO4
max (M-QIDA)emp
SO4
6
Ansatz configurations Ansatz configurations
Figure A3: H2 O 6-31G CAS(4,4) system.
Figure A1: BeH2 INOs system.

0 -67.395 0 -132.607

20 -67.408 20 -132.621

68
58.24
40 -67.421 40 85.22 -132.635
Energy (Ha)

65 64 64 68
Energy (Ha)
ECORR(%)

ECORR(%)

-83.97 54.13 54.1 57.12

41.22 54.95 54.95 82.17
60 -67.434 60 -132.648

80 -67.447 80 66 -132.662
20.86
79.99

100 -67.46 100 -132.675

(L)CX
5 (M-QIDA)SO4
max (M-QIDA)SO4
emp (L)CX
6 (M-QIDA)SO4
max (M-QIDA)SO4
emp
Ansatz configurations Ansatz configurations
Figure A2: NH3 INOs system. Figure A4: N2 cc-PVTZ CAS(6,6) system.

25
 Optimization trajectories for each of the INOs sys-
tem of 50 VQE for HEA ladder-fashion circuit against
Multi-QIDA circuits.
0
0 (L)CX
6
(L)CX All runs

Ecorr(%)
6
All runs 50 Avg
Ecorr(%)

50 Avg Best
Best 100
100 0
0 (M-QIDA)SO4
max
(M-QIDA)SO4

Ecorr(%)
max
50
Ecorr(%)

50
100
100 0
0 (M-QIDA)SO4
emp
(M-QIDA)SO4

Ecorr(%)
emp
50
Ecorr(%)

50
100
100 0 500 1000 1500 2000 2500 3000 3500 4000
0 500 1000 1500 2000 2500 3000 3500 4000 Iterations
Iterations
Figure A7: H2 O 6-31G CAS(4,4) system convergence
Figure A5: BeH2 INOs system convergence trajectories trajectories.

0
(L)CX
6
All runs
Ecorr(%)

50 Avg
Best
100
0
(M-QIDA)SO4
max
Ecorr(%)

50 0
(L)CX
6
All runs
Ecorr(%)

100 50
0 Avg
(M-QIDA)SO4
emp
Best
100
Ecorr(%)

50 0
(M-QIDA)SO4
max
Ecorr(%)

100 50
0 500 1000 1500 2000 2500 3000 3500 4000
Iterations
100
Figure A6: NH3 INOs system convergence trajectories 0
(M-QIDA)SO4
emp
Ecorr(%)

50
Optimization trajectories for each of the
CASCI/Active Region system of 50 VQE for 100
0 1000 2000 3000 4000 5000
HEA ladder-fashion circuit against Multi-QIDA Iterations
circuits.
Figure A8: N2 cc-PVDZ CAS(6,6) system convergence
trajectories.

26