Sample-based quantum diagonalization as parallel fragment solver for the localized active space self-consistent field method
Qiaohong Wang, Mario Motta, Ruhee D'Cunha, Kevin J. Sung, Matthew R. Hermes, Tanvi Gujarati, Yukio Kawashima, Yu-ya Ohnishi, Gavin O. Jones, Laura Gagliardi
physics.chem-phquant-phSubmitted Dec 16, 2025• Announced Dec 18, 2025• 0 citations
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(H$_2$O)$_4$]$_2$bpym$^{4+}$ compound and the [Fe$^{\mathrm{III}}$Fe$^{\mathrm{III}}$Fe$^{\mathrm{II}}$($μ$$_3$-O)-(HCOO)$_6$] complex for calculating the intermediate-spin ground state energies. We observe that LASSQD can tackle fragment sizes intractable by LASSCF, achieves within 1kcal/mol agreement to LASSCF, and delivers results that are competitive with alternative classical methods to solve the Schrödinger equation, and thus can be used as a starting point for a perturbative treatment (LASSQD-PDFT) to recover correlation external to the active space.
Accurately and efficiently describing strongly correlated electronic systems is a central challenge in quantum computational chemistry, with classical and quantum computers. The localized active space self-consistent field method (LASSCF) uses a product of fragment active spaces as a variational space, with the Schrödinger equation solved exactly in each fragment and the fragment active-space orbitals defined in a self-consistent manner. LASSCF is accurate for systems with strong intra-fragment and weak inter-fragment correlation, and its computational cost is combinatorial with respect to the size of the individual fragment active spaces, rather than their product. However, exactly solving the Schrödinger equation in each fragment remains a substantial bottleneck. Here, we address the possibility of solving the fragment active space Schrödinger equation with approximate methods, particularly sample-based quantum diagonalization (SQD). SQD is a technique that uses a quantum computer to sample configurations from a chemically motivated quantum circuit and a classical computer to mitigate errors and solve the Schrödinger equation in a subspace of the configuration space. We apply the proposed method, LASSQD, to the [Fe(H2O)4]2bpym4+ compound and the [FeIIIFeIIIFeII(μ$$_3-O)-(HCOO)6] 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.
We present a quantum-classical algorithm called LAS-QKSD for multireference systems, by combining a classical localized active space (LAS) fragment-based multireference algorithm with the quantum Krylov subspace diagonalization (QKSD) method for quantum computers. The algorithm uses wave-function information from a LAS self-consistent field (LASSCF) calculation to prepare an initial state with better overlap with the target ground state than the Hartree-Fock state. This is coupled with the use of QKSD to ultimately converge to the exact energy, providing faster convergence than starting from the Hartree-Fock state. Fragmentation has the twofold benefit of fewer configurations on the classical side of the algorithm as well as fewer state preparation gates on the quantum side. First, we compare the LAS-QKSD method to the classical LASSCF method and to QKSD with a Hartree-Fock initial state. We then examine ways to load the LASSCF wave function using direct initialization and a QKSD-motivated spectral filtering approach. Finally, using a bimetallic complex, we show that the LAS-QKSD method is an efficient alternative to highly expensive complete active space self-consistent field calculations on strongly correlated systems.
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.
Sampling-based quantum diagonalization method exploits Quantum-centric supercomputing platforms to sample bitstrings for Hamiltonian projection on a quantum computer, and then classically diagonalize the Hamiltonian to estimate the eigenvalues and eigenvectors. In current quantum devices an algorithm is useful when shallow quantum circuits with error mitigation support can discover better results while having either a proof of convergence or some method to explain trust in experiment. In this paper, we introduce DF-SQD, a hybrid algorithm that derives deterministic auxiliary-field circuits from selected double-factorization leaves of the two-electron tensor. The circuits propose occupation-number configurations, while selected configuration interaction evaluates the original active-space Hamiltonian and can recentre subsequent proposal rounds. On N2 (32 qubits; 6-31G basis) and a 40-qubit [Fe2S2(SCH3)4]2- active-space Hamiltonian, we show that DF-SQD improves the energy obtained from sampled determinant spaces while using shallow number-preserving circuits in both simulator and hardware runs. For N2, DF-SQD is 45x more accurate with a 11.23\% smaller subspace, and due to its ability to sample better bitstrings at lesser shots it is 2.93x faster than SQD in quantum devices. For the iron-sulfur cluster, DF-SQD generated a subspace dimension of 221M with 400K shots, while SQD needed 1.5M shots to generate a 238M subspace, thus we have better subspace recovery evident from the hardware at 3.75x reduced shots. At a matched 50M subspace dimension, DF-SQD is 1.32x more accurate (achieves a 24.5\% relative error reduction over standard SQD). So overall, our method is able to discover better results with shallower circuits, is sample efficient, uses configuration recovery (so has targeted error mitigation) and we have empirical convergence observation.
Quantum-chemical simulations are essential for predicting energies of chemical reactions. Accurately solving the many-body Schrödinger equation for reagent and product states of most relevant chemical processes is, however, unfeasible. Quantum computing offers a pathway for predicting energies of correlated electronic systems with localized interactions. Here, we apply a quantum embedding approach for investigating oxygen reduction reactions at the electrode surface in Lithium batteries, a representative example of energetic analysis in localized chemical reactions. We employ an active space selection method based on density difference analysis for identifying the orbitals involved in the reaction. Leveraging the Local Unitary Cluster Jastrow ansatz for state preparation, the active-space orbitals are then processed on a quantum computer. As quantum algorithms, we use Sample-based Quantum Diagonalization, SQD, and its extended version, Ext-SQD, which integrates electronic excitations into the quantum-selected electronic configuration subspace. The largest configurations are represented by quantum circuits mapped onto 80 qubits of an IBM Heron R2 quantum processing unit. For up to 12 orbitals, we are able to benchmark the quantum-computed reaction energies against results obtained with Complete Active Space Configuration Interaction. For benchmarking results in active spaces as large as 32 orbitals, we resort to Heat-Bath Configuration Interaction and Coupled Cluster Singles and Doubles calculations, respectively. At 27 orbitals, the Ext-SQD results exhibit prediction accuracy improvements with regard to the standard, quantum-chemical reference methods that remain computationally feasible at that scale. The results indicate the potential of sample-based quantum diagonalization for performing high-accuracy reaction modeling in chemistry and materials science.
Quantum Selected Configuration Interaction (QSCI) and an extended protocol known as Sample-based Quantum Diagonalization (SQD) have emerged as promising algorithms to solve the electronic Schrödinger equation with noisy quantum computers. In QSCI/SQD a quantum circuit is repeatedly prepared on the quantum device, and measured configurations form a subspace of the many-body Hilbert space in which the Hamiltonian is diagonalized classically. For the dissociation of N 2 and a model [2Fe – 2S] cluster (correlating 10 electrons in 26 orbitals and 30 electrons in 20 orbitals, respectively) we show that a nonperturbative stochastic approach, phaseless auxiliary-field quantum Monte Carlo (ph-AFQMC), using truncated SQD trial wave functions obtained from quantum hardware can recover a substantial amount (e.g., O (100) mHa) of correlation energy. This hybrid quantum-classical combination has the potential to greatly reduce the sampling burden placed on the QSCI/SQD procedure, and is a compelling alternative to recently proposed hybrid ph-AFQMC algorithms that rely on quantum state tomography.