Sample-based quantum diagonalization (SQD), equivalently quantum-selected configuration interaction (QSCI), has in two years become a pragmatic centre of gravity of pre-fault-tolerant quantum chemistry: a quantum processor samples electronic configurations, and the many-electron Hamiltonian is diagonalized classically in the resulting determinant subspace. Its accuracy is set entirely by which configurations enter that subspace, a selection problem for machine learning made acute by a coupon-collector bottleneck. We critically review the ecosystem of generative and learned selectors, organizing it by the object each method generates and the importance signal it exploits, and expose one conspicuous gap: a reward-proportional generative-flow-network proposer built for tail discovery. We then confront the field's central question -- whether the quantum sampler beats classical selected configuration interaction -- and report a carefully scoped negative: across published same-active-space comparisons, strong classical selected CI matches or beats the quantum-sampled subspace, and the flagship single-layer circuits now admit polynomial-time classical energy estimation. We distil a benchmarking standard and turn the negative into a regime map, then test it with FCI-exact experiments that confirm one prediction and refute another: the cheap prior's rank correlation with the exact weights declines with multireference character (a usable coordinate), but a controlled single-molecule noise sweep shows the one generative advantage we find, robustness to valid-shot starvation, to be generic rather than the multireference-specific effect a confounded contrast first suggested. Finally, we flag learning from quantum experiments, whose classical sample-complexity lower bound is an unconditional theorem, as the one adjacent frontier where a quantum advantage is provable but not yet bridged to chemistry.
Comments
30 pages, 10 figures. Review. Code and a reproducible notebook: https://github.com/nicolasbonilla/ml-for-sqd-review
Sample-based quantum diagonalization (SQD), equivalently quantum-selected configuration interaction (QSCI), has in two years become a pragmatic centre of gravity of pre-fault-tolerant quantum chemistry: a quantum processor samples electronic configurations, and the many-electron Hamiltonian is diagonalized classically in the resulting determinant subspace. Its accuracy is set entirely by which configurations enter that subspace, a selection problem for machine learning made acute by a coupon-collector bottleneck. We critically review the ecosystem of generative and learned selectors, organizing it by the object each method generates and the importance signal it exploits, and expose one conspicuous gap: a reward-proportional generative-flow-network proposer built for tail discovery. We then confront the field's central question -- whether the quantum sampler beats classical selected configuration interaction -- and report a carefully scoped negative: across published same-active-space comparisons, strong classical selected CI matches or beats the quantum-sampled subspace, and the flagship single-layer circuits now admit polynomial-time classical energy estimation. We distil a benchmarking standard and turn the negative into a regime map, then test it with FCI-exact experiments that confirm one prediction and refute another: the cheap prior's rank correlation with the exact weights declines with multireference character (a usable coordinate), but a controlled single-molecule noise sweep shows the one generative advantage we find, robustness to valid-shot starvation, to be generic rather than the multireference-specific effect a confounded contrast first suggested. Finally, we flag learning from quantum experiments, whose classical sample-complexity lower bound is an unconditional theorem, as the one adjacent frontier where a quantum advantage is provable but not yet bridged to chemistry.
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.
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.
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.
The energy is the last thing to fail. Sample-based quantum diagonalization (SQD), the method behind the largest chemistry calculations on quantum processors, grades its results by the energy, the quantity least sensitive to errors in the state. This work asks what else the state gets right, on archived hardware data from two sources: the author's published N₂ run on IBM hardware, and IBM's archive for the 77-qubit iron–sulfur demonstration of 2025, the largest SQD calculation whose raw samples are public. Three density properties are followed alongside the energy against converged references: two stop improving while the energy still does; the third, set by the diffuse tail, oscillates while the energy descends smoothly. Four controls show whose work the energy is: random bitstrings seeded with five valid strings lag the hardware data by one iteration and match them at equal subspace size, a coupled-cluster prior matches or beats them there, and the device beats only the published control, uniform in-sector strings, whose loop has nothing to repair. The ordering repeats on the iron–sulfur cluster, where the spin is read alongside the energy and a penalty confirms the classical seed's lead as a singlet energy. Exact samples of the one-layer LUCJ circuit stall on a few hundred configurations: the noise feeds the repair. The recommendation is cheap: report a density property and the spin alongside the energy, at fixed subspace size, against a converged reference. Looking forward, exact samples of a time-evolved state broaden the support ten- to twentyfold over the LUCJ circuit, take the static loop to half a millihartree where LUCJ stalls, and, repaired from a noise channel, lead the hardware shots at equal subspace size at equilibrium and at the stretched bond; generative circuit compression would bring such a sampler inside the Nighthawk gate budget. v2 (2026-09-11): Table 10 gains the one-layer LUCJ circuit with optimized parameters, all-to-all and hardware pattern, as the rung between the coupled-cluster-parametrized circuit and the time-evolved state; §3.6.3 and §3.7.6 state that the configuration prior is coupled cluster's output, not the circuit's input. Changes in v3 (2026-09-13): the optimized-circuit rows of Table 10 restated from four converged optimizations (hardware pattern reproducible to chemical accuracy; three all-to-all optima, a perturbed start returning to one of them); a new §3.6.4 measuring the classical solver's envelope to fifty million determinants on a laptop (Table 12). Changes in v4: Table 10 gains the full-CI weight missing from each sample's string product space, 1 − W(U×U), and two full-CI yardstick rows; §3.6.3 reads it: the missing weight orders the static-loop error where configuration and string counts do not, and a millihartree needs less than about 3 × 10⁻⁴ missing. Changes in v5: a new §3.6.5, "Which root is the state", summarizes in one paragraph a measurement reported in full in a companion paper in preparation: at the stretched geometries the lowest root of a subspace closed under spin inversion is a spin mixture, which a DMRG-X-type overlap rule cannot repair; the abstract and §4 follow it. Table 10's hardware pattern read from the CCSD amplitudes is given as a range over ten builds, because round-off-level differences between coupled-cluster runs change that circuit; three optimizations "from the same starting state".
Abstract We port the Generative Quantum Eigensolver + Quantum-Selected Configuration Interaction (GQE+QSCI) architecture from quantum chemistry to combinatorial graph optimization. Our framework, GQO+QSS (Generative Quantum Optimization + Quantum-Selected Search), follows the same principle: ML generates circuit programs, quantum samplers produce bitstrings, classical recovery repairs and expands them into a restricted subspace, and a classical solver finds the optimum within that subspace. We formalize the key quantity as the recovery-basin mass pr(qθ,G) = Prx∼qθ [x ∈ Ar(G)], where Ar(G) = {x : Er(R(x))∩X∗(G) ̸= ∅} is the preimage of the optimal set under repair R and r-flip expansion Er. Under B independent shots, success probability becomes Psuccess = 1 − (1 − pr)B. The quantum-benefit question reduces to: does pr(qquantum,G) > pr(qclassical,G)? We test this on MaximumWeighted Independent Set (MWIS) with a preregistered, frozen protocol. All 150 ground truths are certified with CBC + brute-force verification. Exact recovery-basin masks are enumerated for all n ≤ 20 instances. Nine samplers are compared on a held-out test set: uniform, greedy, simulated annealing, tabu search, multi-start local search, Metropolis, warm-start QAOA, oracle QAOA, and a matched learned classical policy. Key findings: 1. Tabu search dominates at all sizes (pr = 0.82–0.94), followed by simulated annealing. All quantum samplers and the learned classical policy lose to both at every problem size under the frozen test protocol. 2. Oracle QAOA loses to SA under fair evaluation: With random QAOA configurations (not per-instance optimized), oracle QAOA achieves pr = 0.014 at n = 28 vs SA’s 0.088. The earlier positive result (pr = 0.185 vs 0.130) required per-instance best-of-50 configuration selection—an oracle upper bound, not a deployable method. 3. Learned classical policy matches oracle QAOA: A matched 140K-parameter classical policy achieves pr = 0.005–0.101, comparable to oracle QAOA’s 0.014–0.113. The quantum circuit’s proposal distribution provides no structural advantage over a matched learned classical model in this setting. 4. Recovery-basin masks reveal the phase structure: At r = 2, uniform-sampling Precover drops from 0.975 (n = 16) to 0.775 (n = 20) at B = 200, identifying the regime where sampler quality matters. 5. Hardware robustness without advantage: 136 cores on Piast-Q show 93.4% recovery consistency between QPU and simulator, but no basin-mass advantage. This is a falsification study: Under rigorous, preregistered evaluation with matched learned baselines, shallow QAOA-derived proposal distributions do not outperform matched classical proposal models, and both lose to strong classical baselines (SA, tabu). The earlier positive oracle result was an artifact of per-instance configuration selection. We claim no quantum advantage.
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