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REVIEW 2 major objections 5 minor 37 references

Improved quantum sampling methods for molecular simulations

T0 review · 2 major / 5 minor · reviewed 2026-08-16 · deepseek-v4-flash

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Pith's one-line read Sample-based quantum diagonalization can look successful for purely classical reasons: when the diagonalization subspace is allowed to grow, uniform random sampling matches or beats quantum-backed SQD benchmarks, so fair benchmarking must…

desk verdict The negative benchmarking critique of SQD is solid and important, but the NOCI advantage is oversold: equal diagonalization dimension does not mean equal classical cost, so the 'fixed classical resource budget' claim needs a rewrite. read the letter →

arxiv 2608.11569 v1 pith:4N5GN2PT submitted 2026-08-12 quant-ph physics.chem-phphysics.comp-ph

Connor van Rossum , Jeffery Cohn , Sally Shrapnel , Riddhi Gupta This is my paper · ORCID
classification quant-phphysics.chem-phphysics.comp-ph
keywords sample-basedquantumdiagonalizationquantum-selectedconfigurationinteractionnon-orthogonalelectronicstructurechemistrybenchmarkingsubspacenoiseresilience
verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

The paper argues that a leading quantum-selected configuration interaction method, sample-based quantum diagonalization (SQD), can appear to perform well for reasons that have nothing to do with quantum sampling: uncontrolled growth of the classical diagonalization subspace. When the subspace size is not fixed, classical uniform random sampling reproduces or even outperforms SQD benchmarks, and adding local depolarizing noise improves apparent performance simply by generating more distinct configurations. The paper therefore proposes that any fair SQD benchmark must control the diagonalization size over unique samples. It then introduces a measurement protocol based on non-orthogonal configuration interaction (NOCI), spreading measurements across optimized orbital bases, and reports improved sample efficiency that persists under fixed classical budgets, indicating higher-quality configurations rather than merely more numerous ones.

What carries the argument

The central object is the classical diagonalization subspace of dimension $d$: the number of unique Slater determinants on which the molecular Hamiltonian is projected. Two growth mechanisms drive the critique: spin-product expansion, which splits a measured bitstring into $\alpha$- and $\beta$-spin halfstrings and recombines them (often as a Cartesian product, the default in the reference implementation), and configuration carryover, which appends important samples from previous iterations. The constructive machinery is a NOCI measurement basis: a set of orbital rotations $\{U_m\}$ optimized classically so that each rotated determinant $|\Phi_m\rangle = U_m|\Psi_0\rangle$ lowers the NOCI ground-state energy, with measurements distributed across these bases. Because determinants from different bases are non-orthogonal, the method assembles overlap and Hamiltonian matrices into a generalized eigenvalue problem $\tilde{H}\mathbf{c} = E\tilde{\Gamma}\mathbf{c}$, solved with a generalized Davidson iteration.

What would settle it

Take a published SQD hardware experiment, fix the diagonalization subspace size $d$ to the number of unique sampled configurations at each iteration, and compare the reported ground-state energy with uniform random sampling under the same $d$. If the reported energy error does not increase when local depolarizing noise is added and does not stay at or below the uniform-random value at matched $d$, the central claim is falsified; measuring how $d$ grows across iterations in the reference implementation would also directly test the proposed mechanism.

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Extended reading notes

Core claim

The paper's central claim is that SQD's apparent robustness to noise is largely an artifact of classical post-processing. Reproducing standard N2 benchmarks, the authors find that replacing quantum measurements with uniformly random configurations matches or beats LUCJ-based SQD whenever the diagonalization subspace is allowed to grow through spin-product expansion and configuration recovery. Increasing local depolarizing noise helps only because it increases the diversity of sampled bitstrings, which after correction and expansion enlarges the subspace. Once the diagonalization size $d$ is fixed, noise no longer helps, and noisy LUCJ converges to uniform random sampling from below. The constructive claim is that measuring in a non-orthogonal basis, built classically from NOCI orbital rotations, discovers energy-lowering configurations more efficiently than measuring only in the Hartree–Fock basis, even with $d$ controlled.

Load-bearing premise

The load-bearing premise is that the simulated SQD pipeline matches literature practice: the diagonalization subspace is built from the Cartesian product of alpha- and beta-spin halfstrings (the default in the commonly used reference implementation), and local depolarizing noise on two-qubit gates captures hardware behaviour; if actual demonstrations already control $d$ or use a different spin expansion, the critique may not apply.

Editorial extensions

If this is right

  • SQD results that do not report the diagonalization size $d$ over unique samples cannot be read as evidence of quantum sampling quality, because noise-driven improvements vanish once $d$ is controlled.
  • Uniform random sampling should be a standard control in SQD benchmarking; in a NOCI basis it can outperform LUCJ sampling for correlated systems, exposing ansatz limitations that Hartree–Fock-basis benchmarks hide.
  • Measuring in NOCI bases improves sample efficiency under fixed classical resource budgets, so the benefit is higher-quality configurations rather than a larger search space.
  • The NOCI rotations can be folded into the terminal orbital-rotation layer of LUCJ ansätze, so the multi-basis measurements require no additional quantum circuit depth.
  • The classical cost shifts to non-orthogonal matrix-element evaluation and overlap-matrix storage, with the paper's scaling estimates showing future work must target scalable NOCI basis construction and inter-basis matrix elements.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

  • As an extension, every QSCI benchmark could require reporting $d$ per iteration and matching a uniform-random-sampling control at equal $d$; that single change would reframe several published SQD demonstrations.
  • The NOCI advantage is largest when the underlying ansatz is weakest and at small diagonalization sizes, so measurement-basis engineering looks most useful for near-term noisy hardware rather than as an asymptotic quantum speedup.
  • Because uniform random sampling in the NOCI basis outperforms LUCJ there, a purely classical strongly correlated method suggests itself: classically optimize orbital rotations, sample configurations randomly, and diagonalize, which could be tested against selected CI.
  • The quantitative noise model is local depolarizing noise, so the thresholds would likely shift under readout errors or crosstalk; testing the NOCI protocol under those noise models is a concrete next step.
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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Referee Report

2 major / 5 minor

Summary. This paper addresses benchmarking and improvement of sample-based quantum diagonalization (SQD) for molecular ground-state energies. The authors simulate SQD with LUCJ ansatz states on N2, H8, and H12 and show that when the classical diagonalization subspace is allowed to grow through spin-product expansion and configuration recovery, uniform random sampling can match or exceed noisy SQD, with the apparent improvement tracking the growth of the diagonalization dimension d. They then propose NOCI-SQD, in which measurements are distributed over multiple classically optimized non-orthogonal orbital bases, and report lower energy errors at fixed d for H12 compared with Hartree-Fock-only measurement, while N2 shows only diversity gains without energy improvement. The paper concludes that fair SQD benchmarking must control d and that measurement-basis engineering is a promising direction for improving quantum sampling methods.

Significance. The negative benchmarking result is significant: it provides a concrete mechanism by which reported SQD gains can be classical post-processing artifacts, and it yields a simple, adoptable prescription that fair benchmarking must report and control the diagonalization dimension d. The d-growth analysis across N2, H8, and H12 is consistent, and the uniform-random baseline is an appropriate control. The NOCI measurement idea is original, and the H12 energy improvements at fixed d are encouraging, as is the honest reporting that N2 shows no energy benefit. However, the resource-fairness claim for NOCI is not supported by the benchmarks: the experiments fix only d, while the paper's own cost model indicates that NOCI-SQD spends substantially more classical work per retained configuration. The absence of code and data also limits independent verification, although the methods are described in enough detail to be reproducible in principle.

major comments (2)
  1. [Abstract, Section III, Table I, Appendix E3] The abstract and conclusion claim that NOCI improvements 'persist even under fixed classical resource budgets' and 'after controlling for classical resources required for diagonalization.' The benchmarks in Figs. 4 and 5 fix only the diagonalization dimension d, not the total classical work. Table I and Appendix E3 show that a NOCI-SQD matrix element costs O(N_cdf kappa^3) versus O(1)-O(N_e^2) for standard SQD, and that NOCI-SQD stores dense 2O(d^2) matrices whereas standard SQD stores a typically sparse O(d^2) matrix. Equal d therefore does not imply equal classical cost; a NOCI run at the same d can expend orders of magnitude more classical effort per retained configuration. The conclusion that NOCI configurations are 'higher quality rather than more numerous' is not established by these experiments. Please either compare at equal estimated classical-work budgets (e.g., FLOPs or memory), or explicitly restate the claim as an improvement at fixed diagonalization dimension only.
  2. [Section II, Figs. 2-3, Section V.A, Appendix B] The negative benchmarking claim is framed in the abstract as applying when 'classical resources are not explicitly constrained,' and the paper also asserts that Cartesian-product spin expansion is 'the default setting under which most SQD experiments in literature appear to have been performed using the software package in Ref. [24].' The only support for this literature-level claim is the qiskit-sqd-addon default; no survey of the cited SQD demonstrations is provided. If a substantial fraction of prior demonstrations already controlled d or used a restricted spin expansion, the statement that uniform random sampling reproduces SQD benchmarks would not generalize. Please provide explicit evidence for the default-setting claim, or soften the wording to state that uniform random sampling can reproduce SQD benchmarks when the Cartesian-product expansion is used and d is uncontrolled.
minor comments (5)
  1. [Section V.A, Figs. 2, 3, 5] The text states that ten trials are used per data point, but the figures show no error bars or scatter. Please report standard errors or show individual trial outcomes, as the central quantitative claim is that uniform random sampling matches or exceeds noisy SQD.
  2. [Section II, Fig. 5 caption] The phrase 'improving beyond the NOCI ground-state energy itself' is confusing: since sampled configurations are added to the NOCI subspace, a lower energy than the (M+1)-determinant NOCI energy is expected and is not an anomaly. Consider rewording.
  3. [Section V.A] The claim that Cartesian-product spin expansion is the default in 'most SQD experiments in literature' would be strengthened by a citation or a brief survey of the referenced SQD papers; currently only the software package default is cited.
  4. [Data and code availability] The statement that code and data are 'available upon reasonable request' is weak for a computational benchmarking paper; a public repository or permanent DOI would materially aid reproducibility and independent verification.
  5. [Section V.A, Fig. 2] The noise model is restricted to local depolarizing noise on two-qubit gates. A sentence noting that readout errors and other hardware noise channels may alter the diversity-enhancement effect would help calibrate the claim's applicability to real devices.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: SQD benchmarks are external comparisons against exact FCI/CASCI and uniform-random baselines, and the NOCI advantage is measured with classically optimized basis information supplied to both measurement arms.

full rationale

No circular step meets the evidentiary bar. The central SQD finding—that unconstrained subspace growth lets uniform random sampling reproduce SQD benchmarks—is an empirical simulation result anchored to external exact CASCI/FCI references (Figs. 2-3 and Appendix B), not an identity: energies are obtained by diagonalizing the same molecular Hamiltonian in subspaces whose dimension d is explicitly tracked and controlled. The NOCI claim is benchmarked against exact FCI energies in Figs. 4-5, and the paper controls for the classically optimized NOCI information by seeding both the HF-only and NOCI measurement strategies with the Hartree-Fock configuration in each of the M NOCI bases, so both arms receive the classical NOCI benefit and the marginal difference is the measured configurations. The NOCI basis construction is admittedly Hamiltonian-dependent classical preprocessing (Eqs. C1-C11), but that is the algorithm's mechanism rather than a target-in-the-input reduction, because the validation energies are external exact diagonalization results. The only self-citation is Ref. [29] (Cohn et al.) for compressed double-factorization computational details; CDF is an independently published Hamiltonian representation used as a numerical tool and is not the load-bearing object of the paper's claims, so it does not raise the circularity score. The paper itself flags a genuine non-circular limitation in Section III and Table I: NOCI-SQD has higher per-matrix-element cost O(N_cdf kappa^3) and dense 2O(d^2) storage relative to standard SQD, which weakens the 'fixed classical resource budget' wording; this is a resource-accounting or correctness concern, not a circularity.

Assumptions & free parameters 5 free parameters · 6 assumptions · 0 invented entities

The central benchmarking claim rests on modeling SQD post-processing as in qiskit-sqd-addon and on depolarizing noise as representative. The NOCI claim rests on classical variational basis construction and CDF Hamiltonian accuracy. Hyperparameters M, Ncdf, r0 are chosen by hand. No invented physical entities.

free parameters (5)
  • M (number of NOCI measurement bases) = 15, 30, 45
    Tunable hyperparameter in Eq. (8)/Appendix C; larger M lowers the initial classical NOCI energy, so part of the observed advantage is controlled by this choice.
  • Ncdf (retained compressed double-factorization terms) = Ncdf = Ncho
    Set equal to number of Cholesky vectors in Appendix D; truncation level chosen by hand, affects cost and accuracy of matrix elements.
  • Batch size r0 = 2668 (N2), 9650 (H8)
    Chosen to match average unique configs from 10^6 measurements in Ref [7] for N2; selected per system and affects subspace size.
  • Carryover threshold = 1.0 (no carryover) in main benchmarks; 1e-4 default for comparison
    Set to unity to isolate spin expansion effects; carryover is a known growth mechanism and the choice changes benchmark outcomes.
  • Local depolarizing noise rate lambda = varied per figure
    Chosen to model hardware noise; the paper shows increasing lambda improves uncapped SQD results but worsens capped ones.
assumptions (6)
  • standard math Born rule: measurement outcomes of a quantum state |Psi> are distributed according to |<x|Psi>|^2 in the measured basis.
    Used throughout Section V A to model SQD sampling; not proven in paper.
  • domain assumption Local depolarizing noise on two-qubit gates captures the dominant hardware errors in SQD experiments.
    Used for LUCJ noisy simulations in Figs. 2, 3, 6; realistic hardware has more complex error channels.
  • domain assumption The qiskit-sqd-addon default Cartesian product of alpha and beta half-strings matches how most literature SQD experiments constructed their diagonalization subspaces.
    Quoted in Section V A; the benchmarking critique depends on this being true for the hardware demonstrations cited.
  • ad hoc to paper NOCI variational optimization (Appendix C) converges to orbital rotations that provide chemically relevant measurement bases for the tested molecules.
    No convergence guarantee or error analysis is given; the claimed improvement depends on the optimized bases being useful.
  • domain assumption Compressed double-factorized representation of the Hamiltonian is sufficiently accurate for the benchmark molecules.
    Used in Appendix D/E for NOCI and sampled determinant matrix elements; no convergence study with Ncdf reported.
  • standard math Slater-Condon rules and generalized Wick contractions for non-orthogonal determinants are valid for evaluating matrix elements.
    Standard quantum chemistry; cited to Refs [18-23].

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Pith. "Pith review of Improved quantum sampling methods for molecular simulations." pith.science (2026). https://pith.science/paper/4N5GN2PT

@misc{pith2026260811569,
  author       = {Pith},
  title        = {Pith review of: Improved quantum sampling methods for molecular simulations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/4N5GN2PT}},
  note         = {Machine review of arXiv:2608.11569}
}
read the original abstract

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.

Figures

Figures reproduced from arXiv: 2608.11569 by the authors.

Figure 1
Figure 1. FIG. 1. Low-energy states of a molecule are associated with [PITH_FULL_IMAGE:figures/full_fig_p002_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. SQD for N [PITH_FULL_IMAGE:figures/full_fig_p004_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. Benchmarking SQD for N [PITH_FULL_IMAGE:figures/full_fig_p005_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. SQD using NOCI bases for H [PITH_FULL_IMAGE:figures/full_fig_p006_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. SQD for H [PITH_FULL_IMAGE:figures/full_fig_p007_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Noisy SQD for H [PITH_FULL_IMAGE:figures/full_fig_p014_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Effect of the SQD carryover threshold on ground-state energy estimation starting with 1000 configurations, [PITH_FULL_IMAGE:figures/full_fig_p016_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. SQD for N [PITH_FULL_IMAGE:figures/full_fig_p018_8.png]

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