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REVIEW 3 major objections 6 minor 57 references

Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimizer Methods

T0 review · 3 major / 6 minor · reviewed 2026-08-11 · deepseek-v4-flash

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Pith's one-line read The paper claims that a hybrid optimizer, QN-SPSA+PSR, improves VQE convergence on the transverse Ising model by keeping the cheap stochastic Fubini-Study metric of QN-SPSA while computing the cost gradient exactly with the…

desk verdict Useful hybrid, but the cost-comparison flaw and an overreaching abstract mean the central claims don't hold yet. read the letter →

arxiv 2412.19176 v3 pith:BJIOQFWI submitted 2024-12-26 quant-ph physics.comp-ph

Duc-Truyen Le , Vu-Linh Nguyen , Cong-Ha Nguyen , Quoc-Hung Nguyen , Van-Duy Nguyen This is my paper · ORCID
classification quant-phphysics.comp-ph MSC 81P68 PACS 03.67.Lx
keywords VariationalQuantumEigensolverNaturalGradientParameter-ShiftRuleSimultaneousPerturbationStochasticApproximationTransverseIsingmodelNISQQN-SPSA+PSRAnsatzconstruction
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 claims that a hybrid optimizer, QN-SPSA+PSR, improves the optimization subroutine of the Variational Quantum Eigensolver (VQE) on the transverse Ising model. The method keeps the cheap Fubini-Study metric estimate of QN-SPSA, which costs four quantum-circuit evaluations per update, but replaces the stochastic gradient of the original scheme with the exact Parameter-Shift Rule gradient. On a 12-spin noiseless benchmark with a two-layer RealAmplitudes ansatz, the authors report that it converges more stably and faster than QN-SPSA+SPSA, outperforms COBYLA and Finite Difference, and comes close to the much costlier QN-BDA+PSR. The same study argues that a symmetry-informed RealAmplitudes ansatz with linear entanglement is sufficient for the Ising model. A sympathetic reader would care because the claimed combination targets the main practical bottleneck of VQE on near-term quantum devices: convergence quality per quantum-circuit evaluation.

What carries the argument

The central object is the QN-SPSA+PSR update rule, $\theta_{k+1} = \theta_k - \eta_k \tilde{g}^+(\theta_k) \nabla f(\theta_k)$, where $\tilde{g}$ is the QN-SPSA stochastic approximation to the Fubini-Study metric obtained from four circuit evaluations per iteration and $\nabla f$ is the exact Parameter-Shift Rule gradient. The stability gain comes from replacing the stochastic SPSA gradient of the predecessor with the exact PSR gradient, leaving stochasticity only in the metric. The argument also leans on the symmetry-informed RealAmplitudes ansatz, whose real parameters exploit the transverse Ising model's real eigenstates and total spin-flip symmetry, reducing the parameter count.

What would settle it

Run QN-SPSA+PSR against QN-SPSA+SPSA and COBYLA on a noisy simulator or real device with shot noise, or on a larger lattice and other values of $h$, and report the distribution of final energies over many random seeds; if the stability margin over QN-SPSA+SPSA vanishes or COBYLA wins, the claimed advantage does not generalize.

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Extended reading notes

Core claim

QN-SPSA+PSR is proposed as an extension of QN-SPSA+SPSA: the Fubini-Study metric is approximated stochastically via second-order SPSA using four quantum expectations per iteration, while the outer gradient of the cost function is computed analytically by the Parameter-Shift Rule. This removes the gradient stochasticity that makes QN-SPSA+SPSA unstable, while retaining the metric's low computational cost. Numerical experiments on the 12-spin transverse Ising model at $h=2$ and $J=1$ show it reaching the ground-state energy with relative error declining faster than COBYLA and Finite Difference, closely approaching QN-BDA+PSR's accuracy, and with more stable convergence than QN-SPSA+SPSA. Additional scans over external field and qubit count show the method tracking the exact average ground-state energy alongside QN-BDA+PSR.

Load-bearing premise

The claimed advantage rests on a single noiseless 12-spin transverse Ising benchmark at $h=2$, $J=1$, with a two-layer RealAmplitudes ansatz and only seven samples per stochastic run, with no variance reported; whether the improvement persists across system sizes, fields, ansätze, or under device noise is assumed rather than demonstrated.

Editorial extensions

If this is right

  • VQE on near-term devices could obtain near-natural-gradient convergence at a cost of roughly four quantum evaluations for the metric plus two parameter-shift evaluations per gradient, independent of the number of parameters.
  • The symmetry-guided RealAmplitudes ansatz with linear entanglement is put forward as sufficient for the transverse Ising model, simplifying the circuit without losing accuracy.
  • QN-SPSA+PSR's reported accuracy on average ground-state energies across external fields and qubit numbers positions it as a practical optimizer choice for Ising-type Hamiltonians.
  • The method's per-iteration overhead is small enough that the authors suggest it can be carried over to other variational quantum algorithms, including QAOA and quantum machine learning training.

Reading between the lines

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

  • The paper's noiseless, seven-sample benchmark does not probe shot-noise-dominated or flat-landscape regimes, so the stability advantage may shrink exactly where stochastic gradients usually help.
  • A direct extension would be to swap the SPSA gradient back in only when noise is high, producing a noise-adaptive interpolation between QN-SPSA+SPSA and QN-SPSA+PSR.
  • The symmetry-reduction argument for the ansatz likely transfers to any Hamiltonian that is real and symmetric under a global spin flip, so the same construction could be tested on other spin models.
  • Reporting per-seed convergence curves rather than only averaged relative errors would let practitioners check whether the improvement is consistent or driven by a few runs.
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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

3 major / 6 minor

Summary. The paper studies VQE for the 1D transverse Ising model on up to 12 qubits, comparing classical optimizers (COBYLA, finite differences, SPSA) with quantum natural-gradient-based optimizers (QN-BDA+PSR, QN-SPSA+SPSA) and proposing a hybrid optimizer, QN-SPSA+PSR, which approximates the Fubini-Study metric via 2-SPSA while computing the cost-function gradient exactly via the parameter-shift rule. The authors motivate a RealAmplitudes ansatz from the model's real-representation, local-interaction, and Z2 symmetry properties, and they report that QN-SPSA+PSR converges faster per iteration and more stably than QN-SPSA+SPSA and matches QN-BDA+PSR at lower estimated cost. The numerical evidence is a noiseless benchmark at h=2, J=1, with a fixed two-layer ansatz and seven samples for stochastic methods.

Significance. The proposed QN-SPSA+PSR method is a plausible and simple combination of existing ideas; if its practical advantage were established with a fair resource comparison, it would be a useful addition to the VQE optimizer toolbox. The paper also gives a transparent complexity table (Table I), releases source code, and uses physical symmetries of the transverse Ising model to justify the ansatz, which is a helpful methodological point. However, as presented, the central empirical claim is not yet established: the convergence plots compare methods per iteration rather than per quantum-circuit evaluation, and the stochastic results lack error bars. The significance of the work therefore depends on the outcome of the requested cost-normalized and statistically grounded revision.

major comments (3)
  1. [§IV.B, Figs. 4–5; Table I] The central claim that QN-SPSA+PSR has 'low computational consumption' and outperforms COBYLA is not supported by the reported convergence plots, because the x-axis is iterations rather than quantum-circuit evaluations. From Table I, one QN-SPSA+PSR iteration costs 2p expectation evaluations for the PSR gradient plus 4 for the QN-SPSA metric; for the 12-site, two-layer RealAmplitudes benchmark (p=36) this is about 76 circuit evaluations, versus 6 for QN-SPSA+SPSA and 1 for COBYLA. Figures 4 and 5 should be replotted against a resource measure such as total circuit executions or objective-function evaluations, or the algorithmic claim should be restricted to per-iteration convergence speed rather than total computational cost.
  2. [§IV.B, Figs. 4–7] Stochastic methods are evaluated on only seven samples, and no error bars, seed information, or per-run spread is reported anywhere in the manuscript. Since Section IV.B explicitly states that stochastic methods are 'evaluated based on seven samples,' the qualitative claims that QN-SPSA+PSR is 'more stable' and 'consistently delivered reliable and accurate results' (Figs. 6–7) are unquantified; a statement about stability requires at least variance or confidence intervals over random SPSA perturbations and over random initializations.
  3. [§II.C.2, Eq. (13), and §IV.B] The ansatz layer count L=2 is selected after 'conducting the experimental survey,' and the same benchmark is then used to demonstrate the optimizer comparison. This creates a selection-on-the-benchmark concern: the reported results do not show that L=2 was chosen by a predetermined rule, nor do they show that the optimizer conclusions survive across other L values, system sizes, or field strengths. Please report the layer survey and at least one out-of-sample check (for example, a different N or h) to support the generalization claims made in the conclusion.
minor comments (6)
  1. [Abstract and Conclusion] Phrases such as 'quantum supremacy' and 'potential quantum supremacy' overstate what a noiseless 12-qubit comparative study can establish; replace them with 'potential advantage' or similar calibrated wording.
  2. [Keywords and Section II.A] The keyword 'Ansazt' and the phrase 'Fubini-study metric' should be corrected, and the first sentence of Section II.A has an apparent grammatical issue that should be reworded.
  3. [Fig. 3 caption] Figure 3 is described as 'illustrative' and partly based on 'bias-informed conjectures'; please state explicitly in the caption that this figure is a schematic and not a numerical result, so that it is not mistaken for simulation data.
  4. [Eq. (12) and Eq. (13)] The counting argument leading from 2^{N+1}-2 to 2^{N-1}-1 is compressed; please expand the steps and state explicitly which ansatz parameterization and symmetry constraints are being assumed so that Eq. (13) can be checked.
  5. [References] References [6] and [8] are the same Nature paper and should be consolidated to avoid duplicate citation.
  6. [Data availability] The data availability statement refers to 'datasets' while the linked repository appears to contain code; please clarify whether raw outputs, processed data, or only scripts are archived, and consider using a versioned DOI for reproducibility.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: QN-SPSA+PSR is benchmarked against external exact ground-state energies, and its components are taken from independent prior work.

full rationale

I walked the paper's derivation chain. The proposed QN-SPSA+PSR optimizer is a composition of two previously published estimators: the QN-SPSA Fubini-Study metric approximation (Eqs. 27-32, citing Gacon et al. and Mari et al.) and the parameter-shift-rule gradient (Eq. 18, citing Schuld et al.). Neither component is defined in terms of the paper's own target claim. The central performance evidence is the relative-error convergence against the exact ground-state energy of the transverse Ising model (Figs. 4-7); that external target is not used to fit any parameter of QN-SPSA+PSR. The hyperparameter L=2 is chosen after an 'experimental survey' in Section IV.B on the same Ising benchmark, which is a selection concern, but it is not a reduction of the optimizer's predicted behavior to its inputs by construction, and the same choice is applied across qubit numbers and compared methods. No uniqueness theorem, self-citation chain, or renamed known result carries the argument. The cost-normalization objection, namely that PSR's 2p gradient cost makes per-iteration comparisons potentially unfair when claiming 'low computational consumption,' is a resource-accounting and correctness concern rather than a circularity, because it does not make the claimed outperformance true by definition. Therefore no circular step meets the evidentiary bar, and the score is 0.

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

The central claim rests on standard results (variational principle, parameter-shift rule) and on domain assumptions from prior literature (2-SPSA unbiasedness, RealAmplitudes expressivity for the TIM). The main free choices are the unreported hyperparameters of the optimizer and the data-informed layer count L=2. No new physical entities are introduced.

free parameters (5)
  • Ansatz layer count L = 2
    Selected based on an experimental survey of the same problem (Section II.C.2, Section IV.B). The theoretical bound Eq. (13) suggests larger L for larger N, so L=2 is a data-informed choice.
  • SPSA perturbation size s_k = not reported
    Controls the second-order SPSA estimator of the Fubini-Study metric in Eq. (29). No schedule or value is given.
  • Regularization constant β = not reported
    Added in Eq. (32) to make the QN-SPSA metric estimator invertible; the paper notes a trade-off for large β but gives no value.
  • Learning rate η_k = not reported
    Used in the gradient update Eq. (19); no learning rate or schedule is reported for any optimizer.
  • Number of stochastic samples = 7
    Stochastic methods are evaluated over seven samples; the paper reports no standard deviations or seed information.
assumptions (5)
  • standard math The variational principle guarantees that the expectation value of the Hamiltonian is an upper bound on the ground-state energy.
    Used in Eq. (1) as the objective of VQE.
  • standard math The parameter-shift rule (PSR) exactly evaluates gradients of expectation values for Pauli rotation gates.
    Invoked in Eq. (18) and used as the exact gradient estimator in QN-SPSA+PSR. Proven in cited Ref. [46].
  • domain assumption The 2-SPSA estimator provides an unbiased approximation of the Fubini-Study metric / Hessian.
    Assumed from Refs. [51,52]; used in Eq. (29)-(30) as the metric estimator in QN-SPSA+PSR.
  • domain assumption The RealAmplitudes ansatz with linear entanglement can represent the ground state of the transverse Ising model.
    Based on the model's real Hamiltonian and Z2 symmetry arguments in Section II.C; used for all simulations.
  • domain assumption The regularized pseudo-inverse in Eq. (32) yields a descent direction for natural gradient updates.
    Assumed to hold for the chosen β and for the QN-SPSA metric estimator; no convergence proof is provided.

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Cite this review

Pith. "Pith review of Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimizer Methods." pith.science (2026). https://pith.science/paper/BJIOQFWI

@misc{pith2026241219176,
  author       = {Pith},
  title        = {Pith review of: Variational Quantum Eigensolver: A Comparative Analysis of Classical and Quantum Optimizer Methods},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/BJIOQFWI}},
  note         = {Machine review of arXiv:2412.19176}
}
read the original abstract

In this study, we investigated the Variational Quantum Eigensolver (VQE) application for the Ising model as a testbed model, in which we thoroughly delved into several optimizers, both classical and quantum, and analyzed the extent to which each of these methods would offer a benefit. We then investigated a new combinatorial optimization scheme, termed QN-SPSA+PSR, in which the Fubini-Study metric is approximated within the Quantum Natural Gradient (QN) framework, with its inner gradient estimated by the Simultaneous Perturbation Stochastic Approximation (SPSA), while the outer gradient of the cost function is evaluated exactly by the Parameter-Shift Rule (PSR). The QN-SPSA+PSR method integrates the QN-SPSA computational efficiency with the precise gradient computation of the PSR, improving the stability of QN-SPSA-based and convergence speed per parameter update while maintaining low computational consumption. Our results provide a potential performance improvement in the VQAs' optimization subroutine, even in Quantum Machine Learning's optimization section, and enhance viable paths toward efficient quantum simulations on Noisy Intermediate-Scale Quantum Computing (NISQ) devices. Additionally, we also conducted a detailed study of quantum circuit ansatz structures in order to find the one that would work best with the Ising model and NISQ, in which we utilized the properties of the investigated model.

Figures

Figures reproduced from arXiv: 2412.19176 by the authors.

Figure 1
Figure 1. FIG. 1: Variational Quantum Eigensolver (VQE) Architecture. This schematic representation illustrates the [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2: The two hardware-efficient ansatz investigated [PITH_FULL_IMAGE:figures/full_fig_p006_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3: The illustratively comparative graphics among [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗
Figures from the paper (3 more)
Figure 4
Figure 4. Figure 4: FIG. 4: Comparison of optimization methods for the [PITH_FULL_IMAGE:figures/full_fig_p011_4.png]
Figure 6
Figure 6. Figure 6: FIG. 6: Estimate the average ground state energy with [PITH_FULL_IMAGE:figures/full_fig_p012_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7: Estimate the average ground state energy with [PITH_FULL_IMAGE:figures/full_fig_p012_7.png]

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Reference graph

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Pith tools

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