A previous report (Okuda, Zenodo DOI: 10.5281/zenodo.22469203) identified a discrete failure mode in sample-based quantum diagonalization (SQD): at low shot counts, classical configuration recovery can lock onto a reproducible but incorrect "wrong attractor," and injecting a single correct configuration into the raw sample can trigger a discrete escape from it. That finding was established on a shallow circuit (N2, CAS(6,6), 166 two-qubit gates) where the wrong-attractor structure itself was directly observable. Here we address two questions left open by that work. First, was the original demonstration merely a sample-size artifact rather than evidence that the specific injected configuration mattered? We answer this with a controlled random-injection experiment (Fe2S2, CAS(12,12), 2053 two-qubit gates, 76-80 independent trials per condition): recovery probability rises from 14.5% with no injection to 100.0% with one pre-specified escape configuration, while a random single-configuration injection of the same size produces only 15.0% recovery, and the escape itself is discrete while the residual energy error continues to improve smoothly with further injected configurations. Second, at circuit depths beyond roughly 1000 two-qubit gates, the wrong-attractor structure itself often becomes too noise-dominated to observe directly, so a practical recovery procedure cannot simply look for and inject "the" known wrong-attractor escape configuration. We present such a procedure: a small set of chemically motivated candidate configurations is generated and injected, and the best candidate is selected using one of three answer-free criteria (variational energy, escape rate from a known baseline attractor, or seed-to-seed energy variance), chosen by a cheap upfront classification of the target system. Across 23 independent hardware-data trials spanning two IBM Quantum backends and 21 distinct molecular systems and active-space combinations, this decision rule selected the correct candidate in all 23 trials, including one case improving accuracy by approximately 1000x. We report this as validation of the decision framework across a broad range of hardware and chemical conditions, not as evidence of answer-free prediction for arbitrary, genuinely unknown multi-reference chemistry, and report the framework's limitations candidly. Open-source implementation: https://github.com/okudat9/escape-config-predictor
A previous report (Okuda, Zenodo DOI: 10.5281/zenodo.22469203) identified a discrete failure mode in sample-based quantum diagonalization (SQD): at low shot counts, classical configuration recovery can lock onto a reproducible but incorrect "wrong attractor," and injecting a single correct configuration into the raw sample can trigger a discrete escape from it. That finding was established on a shallow circuit (N2, CAS(6,6), 166 two-qubit gates) where the wrong-attractor structure itself was directly observable. Here we address two questions left open by that work. First, was the original demonstration merely a sample-size artifact rather than evidence that the specific injected configuration mattered? We answer this with a controlled random-injection experiment (Fe2S2, CAS(12,12), 2053 two-qubit gates, 76-80 independent trials per condition): recovery probability rises from 14.5% with no injection to 100.0% with one pre-specified escape configuration, while a random single-configuration injection of the same size produces only 15.0% recovery, and the escape itself is discrete while the residual energy error continues to improve smoothly with further injected configurations. Second, at circuit depths beyond roughly 1000 two-qubit gates, the wrong-attractor structure itself often becomes too noise-dominated to observe directly, so a practical recovery procedure cannot simply look for and inject "the" known wrong-attractor escape configuration. We present such a procedure: a small set of chemically motivated candidate configurations is generated and injected, and the best candidate is selected using one of three answer-free criteria (variational energy, escape rate from a known baseline attractor, or seed-to-seed energy variance), chosen by a cheap upfront classification of the target system. Across 23 independent hardware-data trials spanning two IBM Quantum backends and 21 distinct molecular systems and active-space combinations, this decision rule selected the correct candidate in all 23 trials, including one case improving accuracy by approximately 1000x. We report this as validation of the decision framework across a broad range of hardware and chemical conditions, not as evidence of answer-free prediction for arbitrary, genuinely unknown multi-reference chemistry, and report the framework's limitations candidly. Open-source implementation: https://github.com/okudat9/escape-config-predictor
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".
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) is a hybrid quantum-classical method that replaces variational optimization with a self-consistent recovery loop over QPU samples. Although SQD is considered robust to noisy samples and imperfect classical inputs, its robustness across practical deployment choices has not been systematically analyzed. As a result, shot budgets, qubit layouts, noise mitigation strategies, and the coupled-cluster singles and doubles (CCSD) amplitudes that initialize the ansatz are often chosen without clear empirical guidance. We analyze SQD robustness on IBM Heron hardware across these dimensions. Structured CCSD-amplitude perturbations, including complete zeroing, produce only modest energy shifts from the clean baseline. Differences across layouts and noise-mitigation settings are large in the first recovery iteration but narrow within a few iterations. Accuracy saturates at moderate shot budgets, while very large budgets slightly worsen recovered energies, likely because working-set selection limits the value of additional samples. These results identify where SQD provides genuine deployment robustness and where its limits remain.
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.
Sample-based Quantum Diagonalization (SQD) recovers ground-state energies by classically diagonalizing a Hamiltonian in the subspace spanned by quantum samples, requiring only bitstrings with sufficient ground-state overlap rather than an accurate variational energy. We reveal and exploit this underexplored robustness property: how much non-Clifford and variational expressivity can be removed from the sampling circuit before SQD accuracy degrades? We answer through two complementary compression techniques: gradient-based operator pruning, which discards low-impact excitation operators, and Clifford rounding, which snaps remaining parameters to the nearest Clifford angle. Both of these techniques can be applied to a VQE ansatz on a qubit-reduced Hamiltonian. A systematic ablation study across 21 molecules shows that median SQD error stays within chemical accuracy even at 50\% compression on both axes, while simulation speedup reaches 33×. Hardware validation on 6 molecules on IBM quantum hardware confirms up to 2.8× transpiled-depth reduction with zero loss in SQD accuracy. Our implementation can be found at: https://github.com/zkysfls/cs-vqe-sqd
Sample-based quantum diagonalization (SQD) offers a powerful route to accurate quantum chemistry on noisy intermediate-scale quantum (NISQ) devices by combining quantum sampling with classical diagonalization. Here we introduce HSQD, a novel half-qubit SQD approach that halves the qubit requirement for simulating a chemical system and drastically reduces overall circuit depth and gate counts, suppressing hardware noise. When modeling the dissociation of the nitrogen molecule with a (10e, 26o) active space, HSQD matches the accuracy of SQD on IBM quantum hardware using only half the number of qubits and 40% fewer measurements. We further enhance HSQD with a heat-bath configuration interaction (HCI) inspired selection of the samples, forming HCI-HSQD. This yields sub-millihartree accuracy across the N2 potential energy surface and produces subspaces up to 39% smaller than those from classical HCI, showing a significant improvement in the compactness of the ground-state representation. Finally, we demonstrate the scalability of HCI-HSQD using iron-sulfur clusters, reaching active spaces of up to (54e, 36o) while using only half as many qubits as the original SQD. For these systems, HCI-HSQD reduces SQD energy errors by up to 76% for [2Fe-2S] and 26% for [4Fe-4S], while also reducing subspace sizes, halving measurement requirements, and eliminating expensive post-processing. Together, these results establish half-qubit SQD as a noise-resilient and resource-efficient pathway toward practical quantum advantage in strongly correlated chemistry.