Sample-based quantum diagonalization (SQD/QSCI) diagonalizes the Hamiltonian classically in determinant subspaces selected from quantum-circuit samples. Its iron–sulfur demonstrations underpin claims of utility-scale quantum chemistry, and a recent critique disputes them at matched subspace dimension. Both sides compare energies; neither reports which spin state was computed. Our spin audit uses the exact determinant-space ⟨S²⟩, the invariant spectrum of the S² block over quasi-degenerate manifolds, and higher moments (⟨S⁴⟩ to ⟨S⁸⟩) bounding the spin-sector weights. We apply it to the public [2Fe-2S] and [4Fe-4S] benchmarks, IBM's released qiskit-addon-sqd pipeline and published hardware data. No audited execution returned the named singlet at competitive energy; for published states unavailable to direct audit, the energy-only records do not establish singlet identity. Under the audited HCI protocols, selected-CI growth approaches a spin-contaminated state (⟨S²⟩ ≈ 4.7 on [2Fe-2S]) whose samples keep the contamination. Spin-inversion completion leaves the energy and ⟨S²⟩ of these states unchanged. The released [2Fe-2S] hardware samples give no energy advantage over the released uniform-random control. A scalar mean does not identify these states: two audited arms share ⟨S²⟩ ≈ 2.00, but one is a triplet eigenstate (Var(S²) = 3×10⁻⁶) and the other a 2:1 singlet–quintet mixture whose third moment bounds its triplet weight below 10⁻⁴. At the largest published dimension (5.625×10⁷ determinants, reconstructed here from the released support), the lowest root, 170 mHa below the published energy, has ⟨S²⟩ = 1.3711, unchanged under an 18.6-fold residual tightening to 1.1% of the 0.285 mHa gap. Its first two moments, with Var(S²) = 6.9, exclude any decomposition confined to S ≤ 2. Near-singlet character appears only as a non-stationary superposition, because determinant selection does not commute with S². Energy-only benchmarks therefore do not establish convergence to the named state. We recommend reporting spin moments with every energy and controlling spin during selection.
Sample-based quantum diagonalization (SQD/QSCI) diagonalizes the Hamiltonian classically in determinant subspaces selected from quantum-circuit samples. Its iron–sulfur demonstrations underpin claims of utility-scale quantum chemistry, and a recent critique disputes them at matched subspace dimension. Both sides compare energies; neither reports which spin state was computed. Our spin audit uses the exact determinant-space ⟨S²⟩, the invariant spectrum of the S² block over quasi-degenerate manifolds, and higher moments (⟨S⁴⟩ to ⟨S⁸⟩) bounding the spin-sector weights. We apply it to the public [2Fe-2S] and [4Fe-4S] benchmarks, IBM's released qiskit-addon-sqd pipeline and published hardware data. No audited execution returned the named singlet at competitive energy; for published states unavailable to direct audit, the energy-only records do not establish singlet identity. Under the audited HCI protocols, selected-CI growth approaches a spin-contaminated state (⟨S²⟩ ≈ 4.7 on [2Fe-2S]) whose samples keep the contamination. Spin-inversion completion leaves the energy and ⟨S²⟩ of these states unchanged. The released [2Fe-2S] hardware samples give no energy advantage over the released uniform-random control. A scalar mean does not identify these states: two audited arms share ⟨S²⟩ ≈ 2.00, but one is a triplet eigenstate (Var(S²) = 3×10⁻⁶) and the other a 2:1 singlet–quintet mixture whose third moment bounds its triplet weight below 10⁻⁴. At the largest published dimension (5.625×10⁷ determinants, reconstructed here from the released support), the lowest root, 170 mHa below the published energy, has ⟨S²⟩ = 1.3711, unchanged under an 18.6-fold residual tightening to 1.1% of the 0.285 mHa gap. Its first two moments, with Var(S²) = 6.9, exclude any decomposition confined to S ≤ 2. Near-singlet character appears only as a non-stationary superposition, because determinant selection does not commute with S². Energy-only benchmarks therefore do not establish convergence to the named state. We recommend reporting spin moments with every energy and controlling spin during selection.
Sample-based quantum diagonalization (SQD/QSCI) selects determinant subspaces from quantum-circuit samples for classical diagonalization; its iron–sulfur demonstrations anchor utility-scale quantum-chemistry claims, and a recent critique disputes them at matched subspace dimension. Both sides benchmark on energy-based diagnostics, not spin identity. We introduce a rotation-invariant spin audit: exact determinant-space ⟨S²⟩, invariant S²-Gram spectra over quasi-degenerate manifolds, and higher spin moments (⟨S⁴⟩ through ⟨S⁸⟩) that bound the spin-sector decomposition, and apply it to the public [2Fe-2S] and [4Fe-4S] benchmarks, IBM's shipped qiskit-addon-sqd pipeline, and IBM's published hardware data. No audited execution returned the named singlet at competitive energy; for published states unavailable to direct audit, the energy-only records do not establish singlet identity. Classical selected-CI growth converges to a spin-contaminated attractor (⟨S²⟩ ≈ 4.7 on [2Fe-2S]); samples drawn from it preserve the contamination; spin-inversion completion permits spin symmetry without producing it; and the released [2Fe-2S] hardware samples yield no energy advantage over the uniform-random control. A scalar mean does not identify these states: two audited arms share ⟨S²⟩ ≈ 2.00, one a triplet eigenstate (Var(S²) = 3×10⁻⁶), the other a 2:1 singlet–quintet mixture whose triplet weight its third moment bounds below 10⁻⁴. At the largest published dimension (5.625×10⁷ determinants, reconstructed here from the released support), the numerically stable lowest root lies 170 mHa below the published energy at ⟨S²⟩ = 1.3711 (stable under an 18.6-fold residual tightening to 1.1% of the 0.285 mHa gap), with Var(S²) = 6.9 and first two moments excluding any decomposition confined to S ≤ 2. Near-singlet character survives only as a non-stationary superposition: selection does not commute with S². Energy-only SQD benchmarks therefore do not establish convergence to the named target state; we propose measured spin moments as a mandatory benchmark output, with spin control active rather than passive.
Pre-print. Sample-based quantum diagonalization (SQD) on N₂ and two iron–sulfur clusters, [2Fe-2S] and [4Fe-4S]: a combined loop on IBM quantum processors that aims each circuit from the loop’s current eigenvector, a determinant-level solver with a spin penalty seeded from the quantum determinants, orbital optimization, and an exact N₂ window, each with its classical control; the energy is reported with ⟨S²⟩. Replication package: 10.5281/zenodo.23705888. Files restricted during review.
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.
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.
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.