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arXiv:2501.07231 · Journal of Chemical Theory and Computation

Critical Limitations in Quantum-Selected Configuration Interaction Methods

Peter Reinholdt, Karl Michael Ziems, Erik Rosendahl Kjellgren, Sonia Coriani, Stephan P. A. Sauer, Jacob Kongsted

Published Jun 30, 2025• 8 citations
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Abstract

Quantum Selected Configuration Interaction (QSCI) methods (also known as Sample-based Quantum Diagonalization, SQD) have emerged as promising near-term approaches to solving the electronic Schrödinger equation with quantum computers. In this work, we perform numerical analysis to show that QSCI methods face critical limitations that severely hinder their practical applicability in chemistry. Using the nitrogen molecule and the iron-sulfur cluster [2Fe-2S] as examples, we demonstrate that while QSCI can, in principle, yield high-quality configuration interaction (CI) expansions similar to classical SCI heuristics in some cases, the method struggles with inefficiencies in finding new determinants as sampling repeatedly selects already seen configurations. This inefficiency becomes especially pronounced when targeting high-accuracy results or sampling from an approximate ansatz. In cases where the sampling problem is not present, the resulting CI expansions are less compact than those generated from classical heuristics, rendering QSCI an overall more expensive method. Our findings suggest a significant drawback in QSCI methods when sampling from the ground-state distribution as the inescapable trade-off between finding sufficiently many determinants and generating compact, accurate CI expansions. This ultimately hinders utility in quantum chemistry applications, as QSCI falls behind more efficient classical counterparts.

DOI
10.1021/acs.jctc.5c00375

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Quantum Selected Configuration Interaction (QSCI) methods (also known as Sample-based Quantum Diagonalization, SQD) have emerged as promising near-term approaches to solving the electronic Schrödinger equation with quantum computers. In this work, we perform numerical analysis to show that QSCI methods face critical limitations that severely hinder their practical applicability in chemistry. Using the nitrogen molecule and the iron-sulfur cluster [2Fe-2S] as examples, we demonstrate that while QSCI can, in principle, yield high-quality configuration interaction (CI) expansions similar to classical SCI heuristics in some cases, the method struggles with inefficiencies in finding new determinants as sampling repeatedly selects already seen configurations. This inefficiency becomes especially pronounced when targeting high-accuracy results or sampling from an approximate ansatz. In cases where the sampling problem is not present, the resulting CI expansions are less compact than those generated from classical heuristics, rendering QSCI an overall more expensive method. Our findings suggest a significant drawback in QSCI methods when sampling from the ground-state distribution as the inescapable trade-off between finding sufficiently many determinants and generating compact, accurate CI expansions. This ultimately hinders utility in quantum chemistry applications, as QSCI falls behind more efficient classical counterparts.
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.
Quantum algorithms for selecting a subspace of Hamiltonians to diagonalize have emerged as a promising alternative to variational algorithms in the NISQ era. So far, such algorithms, which include the quantum selected configuration interaction (QSCI) and sample-based quantum diagonalization (SQD), have been formulated within second-quantization in Fock space, which leads to inefficient usage of qubit resources. We introduce the first QSCI algorithm developed in the CI-matrix (CIM) framework, which is known to have optimal qubit scaling of exactly ⌈log⁡2(N)⌉\lceil \log_2 (N) \rceil⌈log2​(N)⌉ where NNN is the size of the CIM. In addition, we introduce a novel single-bit flip error mitigation which comes at the overhead of a single qubit and we combine this with a stochastic approximate Trotterization evolution adapted from qDRIFT. Simulating benchmark N2_22​ and naphthalene molecules on quantum hardware, our results achieved similar accuracy as SQD methods but with significantly less quantum resources. However, our CIM-QSCI algorithm and SQD methods could not match the performance of classical heat-bath CI (HCI) for the same task. Hence, we introduce an augmented version of QSCI called quantum selected heat-bath CI (QSHCI). This variant replaces classical heat-bath sampling with quantum sampling from QSCI to achieve performance comparable to HCI. We note that a current drawback of our approach is the preprocessing cost of O(N2log⁡N)\mathcal{O}(N^2\log N)O(N2logN) for constructing the CIM and performing the Pauli decomposition. This can be further improved by considering efficient CIM access models for the stochastic Trotter evolution.
Quantum-selected configuration interaction (QSCI) methods use a quantum computer to identify dominant electronic configurations in the molecular ground state, while a classical computer diagonalizes the Hamiltonian within the reduced subspace spanned by those configurations. Sample-based quantum diagonalization (SQD), a leading QSCI approach, uses iterative classical post-processing to correct noisy quantum measurement to ensure that the corresponding configurations remain physically sensible. In this work, we show that SQD performance can be strongly influenced by uncontrolled growth of the classical diagonalization subspace. When classical resources are not explicitly constrained, classical uniform random sampling can reproduce SQD benchmarks as noise increases the diversity of sampled configurations. We show any fair benchmarking protocol of SQD must explicitly control diagonalization size over unique samples. We then address the problem of efficiently discovering physically relevant, energy-lowering configurations by introducing a measurement protocol based on non-orthogonal configuration interaction (NOCI). By distributing measurements across orbital bases optimized with respect to the molecular Hamiltonian, we obtain improved sample efficiency relative to measurements performed solely in the Hartree--Fock basis. Importantly, these improvements persist even under fixed classical resource budgets, demonstrating that the resulting configurations are of higher quality rather than being more numerous. Under our proposed benchmarking procedure, we establish measurement-basis engineering as a promising route to improving quantum sampling methods for electronic structure.
Quantum-selected configuration interaction (QSCI) utilizes an input quantum state on a quantum device to select important bases (electron configurations in quantum chemistry) that define a subspace in which to diagonalize a target Hamiltonian, i.e., perform selected configuration interaction, on classical computers. Previous proposals for preparing a good input state, which is crucial for the quality of QSCI, based on optimization of quantum circuits may suffer from optimization difficulty and require many runs of the quantum device. Here, we propose using a time-evolved state by the target Hamiltonian (for some initial state) as an input of QSCI. Our proposal is based on the intuition that the time evolution by the Hamiltonian creates electron excitations of various orders when applied to the initial state. We numerically investigate the accuracy of the energy obtained by the proposed method for quantum chemistry Hamiltonians describing electronic states of small molecules. Numerical results reveal that our method can yield sufficiently accurate ground-state energies for the investigated molecules. Systematic analysis when increasing the number of qubits in a hydrogen chain shows that the subspace size required for sufficiently accurate results is reasonable at system sizes that cannot be solved by naive classical diagonalization. Our proposal provides a systematic and optimization-free method to prepare the input state of QSCI and could contribute to practical applications of quantum computers in quantum chemistry calculations.
Sample-based Quantum Diagonalization (SQD), an extension of Quantum Selected Configuration Interaction (QSCI), has emerged as a promising hybrid quantum-classical paradigm for computing molecular ground state energies. By leveraging quantum sampling instead of variational optimization, QSCI avoids barren plateaus and enables direct reconstruction of correlated electronic wavefunctions. However, existing configuration recovery techniques primarily enforce symmetry constraints without guaranteeing optimal selection of the most physically relevant configurations, often leading to unnecessarily large subspaces and increased classical diagonalization costs. In this work, we introduce a machine-learned compact subspace generation protocol based on Restricted Boltzmann Machines (RBMs), termed QSCI-RBM, and integrate it within the Density Matrix Embedding Theory (DMET) framework. The RBM is trained on quantum-sampled configurations to learn the underlying probability distribution of dominant determinants, enabling the targeted generation of high-probability configurations. We apply this framework to the simulation of a protein-ligand complex involving the inhibitor Carmofur bound to the SARS-CoV-2 main protease (MproM^{\text{pro}}Mpro). Our results demonstrate that DMET-QSCI-RBM achieves energies within the chemical accuracy threshold by accessing only approximately 4% of the configuration subspace. In contrast, standard DMET-SQD simulations failed to reach chemical accuracy while accessing up to 20% of the subspace, even as the chemical potential itself nearly converged. These findings highlight that RBM-assisted configuration generation produces significantly more compact subspaces while preserving physical accuracy, thereby reducing classical computational overhead and enabling the scalable quantum embedding simulation of complex biological systems.
Quantum-selected configuration interaction (QSCI) is a promising hybrid quantum-classical approach in which a quantum device generates configurations for subsequent classical diagonalization. Here, we analyze the performance of QSCI combined with the local unitary cluster Jastrow (LUCJ) ansatz, focusing on the interplay between ansatz expressivity, sampling, noise, and configuration recovery. Using the dissociation of N2 in a large active space as a benchmark, we show that noiseless LUCJ sampling produces compact and biased configurational spaces, limiting the accuracy of the resulting CI energies, particularly in strongly correlated regimes. By introducing a simple noise model, we demonstrate that sampling noise can enhance Hilbert-space exploration by generating additional configurations beyond those supported by the ideal ansatz. When combined with configuration recovery, this leads to systematically improved energies. Moreover, recovery alone (starting from randomly generated configurations) can efficiently construct accurate CI spaces, highlighting its central role in QSCI.