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arXiv:2501.13371 · The Journal of Physical Chemistry A

Nonunitary Variational Quantum Eigensolver with the Localized Active Space Method and Cost Mitigation

Qiaohong Wang, Ruhee D’Cunha, Abhishek Mitra, Stephen K. Gray, Matthew Otten, Laura Gagliardi

Published Aug 18, 2025• 3 citations
arXivHTMLPDF

Abstract

Accurately describing strongly correlated systems with affordable quantum resources remains a central challenge for quantum chemistry applications on near and intermediate term quantum computers. The localized active space self-consistent field (LASSCF) approximates the complete active space self-consistent field (CASSCF) by generating active space-based wave functions within specific fragments while treating interfragment correlation with mean-field approach, hence is computationally less expensive. Hardware-efficient ansatzes (HEA) offer affordable and shallower circuits, yet they often fail to capture the necessary correlation. Previously, Jastrow-factor-inspired nonunitary qubit operators were proposed to use with HEA for variational quantum eigensolver (VQE) calculations (so-called nuVQE), as they do not increase circuit depths and recover correlation beyond the mean-field level for Hartree–Fock initial states. Here, we explore running nuVQE with LASSCF as the initial state. The method, named LAS-nuVQE, is shown to recover interfragment correlations, reach chemical accuracy with a small number of gates (<70) in both H 4 and square cyclobutadiene (C 4 H 4 ), and produces more accurate energetics than its HEA counterparts at all circuit depths. To further address the inherent symmetry-breaking in HEA, we implemented spin-constrained LAS-nuVQE to extend the capabilities of HEA further and show spin-pure results for square cyclobutadiene. We also mitigate the increased measurement overhead of nuVQE via Pauli grouping and shot-frugal sampling, reducing measurement costs by up to 2 orders of magnitude compared to ungrouped operator, and show that one can achieve better accuracy with a small number of shots (10 3–4 ) per one expectation value calculation compared to noiseless simulations with one or two orders of magnitude more shots. Finally, wall clock time estimates show that, with our measurement mitigation protocols, nuVQE becomes a cheaper and more accurate alternative than vanilla VQE with HEA. Taken together, these developments illustrate a practical pathway toward performing multireference chemical simulations with accuracy and affordable resources on today’s quantum hardware, achieving both accuracy and affordability in challenging correlated systems.

DOI
10.1021/acs.jpca.5c03028

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Accurately describing strongly correlated systems with affordable quantum resources remains a central challenge for quantum chemistry applications on near and intermediate term quantum computers. The localized active space self-consistent field (LASSCF) approximates the complete active space self-consistent field (CASSCF) by generating active space-based wave functions within specific fragments while treating interfragment correlation with mean-field approach, hence is computationally less expensive. Hardware-efficient ansatzes (HEA) offer affordable and shallower circuits, yet they often fail to capture the necessary correlation. Previously, Jastrow-factor-inspired nonunitary qubit operators were proposed to use with HEA for variational quantum eigensolver (VQE) calculations (so-called nuVQE), as they do not increase circuit depths and recover correlation beyond the mean-field level for Hartree–Fock initial states. Here, we explore running nuVQE with LASSCF as the initial state. The method, named LAS-nuVQE, is shown to recover interfragment correlations, reach chemical accuracy with a small number of gates (<70) in both H 4 and square cyclobutadiene (C 4 H 4 ), and produces more accurate energetics than its HEA counterparts at all circuit depths. To further address the inherent symmetry-breaking in HEA, we implemented spin-constrained LAS-nuVQE to extend the capabilities of HEA further and show spin-pure results for square cyclobutadiene. We also mitigate the increased measurement overhead of nuVQE via Pauli grouping and shot-frugal sampling, reducing measurement costs by up to 2 orders of magnitude compared to ungrouped operator, and show that one can achieve better accuracy with a small number of shots (10 3–4 ) per one expectation value calculation compared to noiseless simulations with one or two orders of magnitude more shots. Finally, wall clock time estimates show that, with our measurement mitigation protocols, nuVQE becomes a cheaper and more accurate alternative than vanilla VQE with HEA. Taken together, these developments illustrate a practical pathway toward performing multireference chemical simulations with accuracy and affordable resources on today’s quantum hardware, achieving both accuracy and affordability in challenging correlated systems.
We propose a modification of the Variational Quantum Eigensolver algorithm for electronic structure optimization using quantum computers, named nonunitary Variational Quantum Eigensolver (nu-VQE), in which a nonunitary operator is combined with the original system Hamiltonian leading to a new variational problem with a simplified wave function ansatz. In the present work, as nonunitary operator, we use the Jastrow factor, inspired from classical Quantum Monte Carlo techniques for simulation of strongly correlated electrons. The method is applied to prototypical molecular Hamiltonians for which we obtain accurate ground-state energies with shallower circuits, at the cost of an increased number of measurements. Finally, we also show that this method achieves an important error mitigation effect that drastically improves the quality of the results for VQE optimizations on today's noisy quantum computers. The absolute error in the calculated energy within our scheme is 1 order of magnitude smaller than the corresponding result using traditional VQE methods, with the same circuit depth.
Accurately and efficiently describing strongly correlated electronic systems is a central challenge in quantum computational chemistry, with classical and quantum computers. The localized active space self-consistent field method (LASSCF) uses a product of fragment active spaces as a variational space, with the Schrödinger equation solved exactly in each fragment and the fragment active-space orbitals defined in a self-consistent manner. LASSCF is accurate for systems with strong intra-fragment and weak inter-fragment correlation, and its computational cost is combinatorial with respect to the size of the individual fragment active spaces, rather than their product. However, exactly solving the Schrödinger equation in each fragment remains a substantial bottleneck. Here, we address the possibility of solving the fragment active space Schrödinger equation with approximate methods, particularly sample-based quantum diagonalization (SQD). SQD is a technique that uses a quantum computer to sample configurations from a chemically motivated quantum circuit and a classical computer to mitigate errors and solve the Schrödinger equation in a subspace of the configuration space. We apply the proposed method, LASSQD, to the [Fe(H2_22​O)4_44​]2_22​bpym4+^{4+}4+ compound and the [FeIII^{\mathrm{III}}IIIFeIII^{\mathrm{III}}IIIFeII^{\mathrm{II}}II(μμμ3_33​-O)-(HCOO)6_66​] complex for calculating the intermediate-spin ground state energies. We observe that LASSQD can tackle fragment sizes intractable by LASSCF, achieves within 1kcal/mol agreement to LASSCF, and delivers results that are competitive with alternative classical methods to solve the Schrödinger equation, and thus can be used as a starting point for a perturbative treatment (LASSQD-PDFT) to recover correlation external to the active space.
We develop a hybrid quantum-classical workflow combining sample-based quantum diagonalization (SQD) and the localized active space self-consistent field method (LASSCF) to solve for the ground states of transition-metal complexes, a longstanding challenge for both classical and quantum algorithms. The resulting approach, named LASSQD, integrates quantum sampling with fragment-based multireference theory to reduce the computational cost of solving strongly correlated active spaces. We test LASSQD on multiple iron-based complexes and demonstrate that it agrees with LASSCF within 1 kcal/mol, albeit at a much reduced computational cost. The cost reduction originates from the use of a sparse approximation of the exact and combinatorially large ground-state wavefunction, which also enables LASSQD to treat fragment sizes that are computationally inaccessible to LASSCF, as demonstrated by our computation of the spin gap of iron-porphyrin. These results establish that LASSQD is a scalable strategy for generating reliable multireference wave functions, providing a robust starting point for post-SCF correlation methods that recover dynamic correlation beyond the active space, and a promising pathway toward quantum-enhanced electronic structure calculations.
High Resolution Image Download MS PowerPoint Slide The variational quantum eigensolver (VQE) algorithm has positioned quantum chemistry calculation methods as promising applications for noisy intermediate-scale quantum (NISQ) devices. In NISQ devices, where the quantum noise is considerable, it is desirable to have an ansatz for the quantum circuit that is practical to implement and capable of achieving high chemical accuracy. Hardware-efficient Ansätze (HEA), such as the RyRz linear ansatz (RLA) and its modified form that preserves physical quantities─the symmetry preserving ansatz (SPA)─are constructed using quantum gates that can be implemented easily on a quantum computer. However, the extent to which HEA can accurately yield the electronic state energies of actual molecules and the number of quantum gates required to achieve chemical accuracy─which, in practical cases, should be within 1 kcal/mol of the exact energy value─are not well understood. In this study, we aimed to gain a detailed understanding of these aspects by performing noiseless simulations to obtain the ground and low-lying excited state using high-depth HEA quantum circuits for several molecules, including LiH, H 2 O, BeH 2, CH 4, and N 2 . From the results, we demonstrated that symmetry preserving HEA, such as SPA, can achieve accurate computational results that maintain CCSD-level chemical accuracy by increasing the number of layers. We quantitatively analyze how the expressibility and entangling capability of RLA and SPA quantum circuits evolve with circuit depth. Our results reveal that while increased layers generally expand accessible Hilbert space and entanglement, differences in their limitations highlight the importance of considering physically allowed entanglement for performance of a parametric quantum circuit in VQE. We also studied the potential energy surface of the dissociation of these molecules and found that quantum circuits using SPA can capture static electron correlation, which is challenging to address with classical single-reference quantum chemistry methods such as CCSD. These results from SPA demonstrate its potential to represent electronic states, as it achieves highly accurate results with fewer gate operations compared to physically inspired quantum circuits such as the Unitary Coupled Cluster (UCC) method. This study establishes a possible path for the application of quantum devices in solving quantum chemistry problems.
Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today's quantum devices. Here we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground state energy. Our algorithm, termed "local active space unitary coupled cluster" (LAS-UCC), scales linearly with system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2_22​)2_22​ into two H2_22​ molecules and by breaking the two double bonds in trans-butadiene and resources estimates are provided for linear chains of up to 20 H2_22​ molecules.
Quantum chemistry calculations of large, strongly correlated systems are typically limited by the computation cost that scales exponentially with the size of the system. Quantum algorithms, designed specifically for quantum computers, can alleviate this, but the resources required are still too large for today’s quantum devices. Here we present a quantum algorithm that combines a localization of multireference wave functions of chemical systems with quantum phase estimation (QPE) and variational unitary coupled cluster singles and doubles (UCCSD) to compute their ground state energy. Our algorithm, termed “local active space unitary coupled cluster” (LAS-UCC), scales linearly with system size for certain geometries, providing a polynomial reduction in the total number of gates compared with QPE, while providing accuracy above that of the variational quantum eigensolver using the UCCSD ansatz and also above that of the classical local active space self-consistent field. The accuracy of LAS-UCC is demonstrated by dissociating (H2)2 into two H2 molecules and by breaking the two double bonds in trans-butadiene and resources estimates are provided for linear chains of up to 20 H2 molecules.