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 ($M^{\text{pro}}$). 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.
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 (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.
Calculation of binding energies for protein-ligand molecular systems requires accurate treatment of the electronic structure, a quantum chemistry problem that scales exponentially on classical hardware, while current quantum hardware remains too noisy for the required circuit depths. This report presents a hybrid quantum-classical workflow performed on the Fujitsu FX700 ideal state-vector simulator using QARP that addresses two structural inefficiencies in quantum-sampling-based diagonalization workflows. First, we integrate the Linear Scaling CNOT UCCSD (LCNot-UCCSD) ansatz into the QSCI framework, replacing the O(N6) CCSD parameter initialization of the competing LUCJ ansatz approach with O(N4) MP2-amplitude initialization. Second, we introduce QSCI-RBM, a variant that replaces the configuration recovery of the SQD framework with a Restricted Boltzmann Machine (RBM) acting as a compact generative subspace expansion model. Both are evaluated on eight different molecules in STO-3G across 14 controlled artificial error levels with 100 independent runs each, validated on potential energy surface scans of the N2 molecule in cc-pVDZ, and embedded within DMET to treat the FDA-approved antiviral Amantadine (C10H17N, 11 DMET fragments) and the active region of the SARS-CoV-2 main protease complexed with its covalent inhibitor Carmofur (PDB: 7BUY, C15H28N4O5S, 10 fragments). To our knowledge, this is the first deployment of LCNot-UCCSD within QSCI on a quantum computing simulator, and the first DMET-QSCI(LCNot-UCCSD)-RBM application to an industry-relevant protein-ligand system. By utilizing a fraction of the classical computing resources required by the current state-of-the-art work by Cleveland Clinic, RIKEN, and IBM Quantum, this approach enables more efficient and economical drug discovery simulations for the industry.
The accurate treatment of electron correlation in extended molecular systems remains computationally challenging using classical electronic structure methods. Hybrid quantum-classical algorithms offer a potential route to overcome these limitations; however, their practical deployment on existing quantum computers requires strategies that both reduce problem size and mitigate hardware noise. In this work, we investigate ground-state energy calculations of ligand-like molecules using Sample-based Quantum Diagonalization (SQD) within the Density Matrix Embedding Theory (DMET) framework, focusing on low-symmetry systems with diverse bonding motifs that exhibit subsystem-dependent variations in fragment-environment entanglement. These entanglement-based variations directly influence bath orbital construction, impurity sizes, and the structure of the embedded Hamiltonians, posing nontrivial challenges for both embedding and quantum sampling. By combining DMET fragmentation with SQD-based construction of reduced configuration spaces through quantum sampling and iterative configuration recovery, we perform quantum simulations on IBM's Eagle R3 (IBM Sherbrooke) and IBM's Heron R3 (IBM Boston) superconducting quantum hardware thereby, showing that the entanglement structure across embedding subsystems plays a central role in determining the efficiency and accuracy of the simulations. Despite these complexities, we show that the DMET-SQD framework yields ground-state energies in strong agreement with DMET-FCI benchmarks, achieving chemical accuracy (1 kcal/mol) across all systems studied. These results demonstrate that SQD-based quantum simulations can be robustly extended to low-symmetry, chemically realistic, industry relevant molecules, and highlight the importance of entanglement-aware embedding strategies for scalable quantum electronic structure calculations.
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.
The Variational Quantum Eigensolver (VQE) and associated hybrid quantum-classical algorithms have long been constrained by the practical limitations of near-term quantum hardware, particularly the susceptibility of classical optimization loops to barren plateaus, circuit depth requirements, and measurement overheads. This research explores "optimization-free" input preparation frameworks for Quantum-Selected Configuration Interaction (QSCI)—also known as Sample-based Quantum Diagonalization (SQD). By replacing parameterized variational circuits with deterministic alternatives such as Real-Time Evolution (TE-QSCI), Classically-Boosted VQE (CB-VQE), and Variational Quantum Imaginary Time Evolution (VarQITE), the burden on quantum hardware is shifted exclusively toward probability sampling. This study demonstrates that these frameworks effectively decouple input state preparation from variational optimization, enabling the generation of compact, highly correlated wavefunctions while retaining the rigorous variational bound guaranteed by classical diagonalization. The integration of stochastic circuit compilation (qDRIFT) and Multireference Perturbation Theory (MRPT) further resolves sampling inefficiencies, offering a noise-resilient pathway for achieving chemical accuracy in large-scale molecular simulations. Also available on Academia.edu: https://www.academia.edu/attachments/133434598/download_file?s=portfolio
Sample-based quantum diagonalization (SQD), equivalently quantum-selected configuration interaction (QSCI), has become a centre of gravity of pre-fault-tolerant quantum chemistry: a processor samples electronic configurations and the Hamiltonian is diagonalized classically in the resulting subspace. Accuracy is governed entirely by which configurations enter it -- a machine-learning selection problem, made acute by a coupon-collector bottleneck. We review the generative and learned selectors by what each generates and the signal it exploits, and identify one gap: no reward-proportional generative-flow-network proposer has been built for tail discovery. On the field's central question -- whether the quantum sampler beats classical selected CI -- the negative verdict is not ours to claim: priority belongs to Reinholdt et al. [JCTC 21, 6811 (2025)], and polynomial-time classical estimation of the flagship circuits has reinforced it. We state that verdict at the precision a falsifiable claim requires -- it concerns reproducible, same-active-space comparisons on molecular electronic structure -- and weigh the claims outside those qualifiers. We show that alpha-string weights are not invariant under rotations inside degenerate orbital shells, so determinant counts are undefined until the orbital gauge is declared. We distil a ten-element benchmarking standard and apply it to our own deposit, which returned defects that changed numbers printed here and retracted one from v1. FCI-exact experiments confirm one prediction and refute another: the single generative advantage we find keeps no consistent sign along the dissociation coordinate at device-calibrated noise and reverses under a symmetric readout model. It does beat a noise-matched classical recovery loop on N2 by a margin five seeds cannot resolve, and loses by over a factor of two to a classical selector that needs no sampler.
The pursuit of exact solutions to the many-body electronic Schrödinger equation is constrained by the exponential scaling of the Hilbert space. While Quantum-Selected Configuration Interaction (QSCI) provides a robust, hybrid quantum-classical alternative to standard Variational Quantum Eigensolvers, it is hampered by significant sampling inefficiencies and finite-shot bottlenecks in strongly correlated regimes. This report analyzes the integration of generative machine-learning models—including autoregressive transformers, Neural-Network Quantum States (NQS), and discrete diffusion models—as a transformative approach to the QSCI paradigm. By treating classical diagonalization outputs as high-quality training data, these generative architectures internalize complex electron correlation patterns, enabling active configuration discovery, dramatic wavefunction compression, and optimized quantum circuit design. This synthesis provides a scalable blueprint for achieving chemical accuracy in KSCCN Mamun, SM (2026) ………………………………..1 Integration of Generative Machine-Learning Models with Quantum-Selected Configuration Interaction: An Academic Extension K S C electronic structure simulations while maintaining the strict variational upper-bound guarantee of classical diagonalization. KSCCN Mamun, SM (2026) ………………………………..2 Integration of Generative Machine-Learning Models with Quantum-Selected Configuration Interaction: An Academic Extension Also available on Academia.edu: https://www.academia.edu/attachments/133495242/download_file?s=portfolio