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REVIEW 3 major objections 4 minor 55 references

Benchmarking a wide range of optimisers for solving the Fermi-Hubbard model using the variational quantum eigensolver

T0 review · 3 major / 4 minor · reviewed 2026-08-12 · deepseek-v4-flash

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Pith's one-line read Across 372 Fermi-Hubbard VQE instances, Momentum and Adam with finite-difference gradients reach the lowest ground-state energies, while SPSA, CMAES, and BayesMGD win when cost-function calls are scarce.

desk verdict Broad, useful VQE optimizer benchmark, but unequal evaluation effort across optimizers weakens the headline ranking. read the letter →

arxiv 2411.13742 v2 pith:JOMEX4FO submitted 2024-11-20 quant-ph cs.LGcs.NE

Benjamin D.M. Jones , Lana Mineh , Ashley Montanaro This is my paper · ORCID
classification quant-phcs.LGcs.NE
keywords variationalquantumeigensolverFermi-HubbardmodelHamiltonianansatzoptimiserbenchmarkingfinitedifferencegradientsimultaneousperturbationnaturalhyperparametersweeping
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

This paper tries to establish which classical optimisers a practitioner should choose when running the variational quantum eigensolver on the Fermi-Hubbard model with the Hamiltonian variational ansatz. It benchmarks 30 optimisers on 372 instances that vary grid size, Coulomb interaction, filling, ansatz depth, and number of measurement shots. Its central finding is that no single optimiser wins on both metrics: Momentum and Adam with finite-difference gradients give the best final accuracy, whereas SPSA, CMAES, and BayesMGD reach a target accuracy in the fewest cost-function calls. The paper also shows that the finite-difference step size strongly controls gradient quality and that simultaneous-perturbation gradients trade final precision for speed. The practical payoff would be concrete guidance for near-term quantum hardware, where the number of circuit evaluations is the scarce resource.

What carries the argument

The central object is the benchmark itself: a fixed set of 372 VQE instances built from the Hamiltonian variational ansatz for the Fermi-Hubbard model, each evaluated with two shot counts (1,000 and 10,000), with exact ground energies computed classically as the reference. The two gradient subroutines are the load-bearing comparison: finite differences, which costs two cost-function evaluations per parameter and is run at step size 0.4 (the average of the best step sizes found by sweeping four instances), and simultaneous perturbation, which costs two evaluations total and is run at step size 0.15. The paper also uses the diagonal approximation of the quantum Fisher information matrix as the machinery for the quantum natural gradient comparison, restricted to one-dimensional systems because the HV ansatz's fixed FSWAP gates prevent a global parameter shift of shared-angle gates.

What would settle it

Run the same 372-instance benchmark with finite-difference step sizes of, say, 0.1 and 0.8 while keeping all hyperparameter sweeps fixed; if the top optimisers in the final-energy ranking shift away from Momentum and Adam, the central ranking is an artefact of the 0.4 step size chosen from four instances.

Watch

Extended reading notes

Core claim

The paper claims that for VQE on the Fermi-Hubbard model with the Hamiltonian variational ansatz under shot-noise-limited cost functions, optimisation performance separates cleanly into two regimes. On final energy accuracy, Momentum and Adam using a finite-difference gradient with step size 0.4 outperform the other optimisers across the 372 instances, with the advantage clearest at 10,000 shots and half filling. On efficiency, SPSA, CMAES, and BayesMGD need the fewest cost-function calls to reach a given tolerance, and CMAES offers the best balance between call count and final energy. The paper further claims that finite-difference gradients give more precise final energies but many more calls than simultaneous-perturbation gradients, and that the quantum natural gradient's per-iteration advantage over vanilla gradient descent is generally erased when counting the extra calls needed to estimate the Fisher information.

Load-bearing premise

The finite-difference step size 0.4, averaged from the best step sizes found on only four sweeping instances, is assumed to work well across all 372 instances; if the optimal step size varies by instance, the ranking of gradient-based optimisers could change.

Editorial extensions

If this is right

  • A practitioner prioritising final ground-state energy should pick Momentum or Adam with a finite-difference gradient at a carefully chosen step size rather than a black-box optimiser.
  • A practitioner with a strict budget on circuit evaluations should pick SPSA, CMAES, or BayesMGD, with CMAES as the best compromise between call count and final energy.
  • Using simultaneous perturbation as a gradient subroutine cuts the number of calls but leaves the run further from the ground state than finite differences; switching from simultaneous perturbation to finite differences partway through a run is a plausible way to combine both advantages.
  • The step size for finite-difference gradients must be treated as a serious hyperparameter: an arbitrary small value such as 0.01 can make all gradient-based optimisers perform poorly.
  • For 1D Fermi-Hubbard systems, the quantum natural gradient and imaginary time evolution can improve per-iteration convergence, but their overhead in cost-function calls means they do not beat vanilla gradient descent when total calls are counted.

Reading between the lines

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

  • Beyond the paper's stated scope, the same two-regime pattern likely appears for other lattice Hamiltonians such as the transverse-field Ising model, because the Hamiltonian variational ansatz keeps parameter counts low and the noise structure is similar; this is a testable transfer, not a paper claim.
  • The step-size averaging over four instances suggests a cheaper protocol: run a small hyperparameter sweep on one or two representative instances before a hardware campaign, then fix the step size for the rest; the paper does not propose this as a protocol.
  • A testable extension is an adaptive schedule that begins with simultaneous-perturbation gradients to descend quickly and switches to finite differences near the end; the paper mentions this as a future idea, and the data here suggest it should beat either subroutine used alone.
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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 / 4 minor

Summary. The paper reports a large numerical benchmark of 30 optimisers applied to 372 instances of VQE for the Fermi-Hubbard model using the Hamiltonian variational ansatz. The authors record both exact and shot-noise cost functions, rank optimisers by final energy and by number of cost-function calls to reach fixed tolerances, analyse finite-difference versus simultaneous-perturbation gradients, and study the quantum natural gradient and imaginary-time evolution for 1D systems. The main claims are that Momentum and Adam with finite-difference gradients give the best final accuracy, that SPSA, CMAES, and BayesMGD are best in the low-call regime, and that the finite-difference step size has a strong effect on performance. Hyperparameters are selected by sweeping on four instances, and the full dataset is made available on Zenodo.

Significance. If the reported ranking is robust, the paper would be a useful reference for VQE practitioners, bringing together a wider set of optimisers, instances, and per-call data than most prior comparisons. The careful recording of every cost-function call, the use of exact ground states as baselines, the open data, and the detailed per-optimiser hyperparameter notes are genuine strengths that make the study reproducible in principle. However, the headline ranking is currently vulnerable to a protocol issue: tuned optimisers are evaluated as the best of up to five independent noisy runs, while zero-hyperparameter scipy optimisers are evaluated once. Because the cost function is stochastic, this unequal number of attempts can inflate the apparent gap between the top optimisers and the scipy baselines. The central conclusions are defensible but need additional analysis to control for this effect.

major comments (3)
  1. [Section 2.5 and Fig. 6 caption] The headline ranking compares best-of-multiple-runs for tuned optimisers against a single run for the zero-hyperparameter scipy optimisers. Section 2.5 states that each tuned optimiser is run with at most five sets of hyperparameters (the default plus the best found on each of the four sweeping instances), and the caption of Fig. 6 says that for each optimiser the reported value is 'the best final energy (over different runs, e.g. for multiple hyperparameter settings)'. Since the cost function is stochastic with 1,000 or 10,000 shots, the minimum of five noisy runs is a biased optimistic estimator, whereas COBYLA, Powell, BFGS, L-BFGS-B, TNC, SLSQP, CG, and Newton-CG contribute only a single run per instance. The gaps in Figs. 6, 7, and 9 between the top tuned optimisers and the scipy optimisers could therefore be inflated by the evaluation protocol rather than reflecting genuine optimizer quality. I recommend adding a control that uses only the default hyperparameter set for every optimiser, and/or giving every optimiser the same number of independent restarts, and reporting both the best-of and the median performance.
  2. [Section 2.4 and Section 2.5] The global finite-difference step size of 0.4 is the average of the best step sizes found on only four instances (0.417, 0.251, 0.324, and 0.550), and the per-optimiser hyperparameters are selected on those same four instances. These sweeping instances are all small (at most 3 × 2 grids and five layers) and do not include the 14-18 qubit or 8-10 layer cases in Benchmarks 3 and 4. Since Fig. 18 shows that optimiser performance varies systematically with U, filling, and number of layers, the representativeness of the four sweeping instances is an unverified assumption that is load-bearing for the claim that '0.4 seemed to be a good choice'. I ask the authors to quantify the sensitivity of the rankings to this choice, for example by re-running the gradient-based optimisers with step sizes near the ends of the observed range (e.g. 0.25 and 0.55) and reporting whether the leading optimisers remain the same.
  3. [Section 3.2.2 and Fig. 9] For the number-of-calls metric it is not stated whether the reported 'calls to reach tolerance' is taken from the best run over the multiple hyperparameter sets or from each run individually. If Fig. 9 uses the best run for each tuned optimiser and the single run for scipy optimisers, the same best-of-multiple-runs bias applies to the low-call ranking as to the final-energy ranking. The authors should state explicitly how the call counts are aggregated and should report both the best-run and the median-run call counts, so that readers can separate genuine optimizer quality from the number of attempts allowed.
minor comments (4)
  1. [Table 1] In the row for '3 × 2', the expression '2m⌈ n−1 2 ⌋' mixes a ceiling bracket with a floor symbol; it should read '2m⌈(n−1)/2⌉'.
  2. [Section 3.2, first paragraph] The phrase 'some plots of individual runs are shown n Fig. 5' should read 'shown in Fig. 5'.
  3. [Table 15 caption] The caption says 'Coordinate descent hyperparameters', but the table lists Nelder-Mead hyperparameters; the caption should be corrected.
  4. [Appendix C.4] The word 'Bernouilli' in the SPSA pseudocode description should be spelled 'Bernoulli'.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the paper reports directly measured benchmark outcomes, with self-citations only supplying the ansatz and algorithm choices, not the ranking.

full rationale

This is an empirical benchmarking study rather than a derivation, and I find no circular step that makes any reported result equivalent to its inputs by construction. The central claims, such as "Momentum or Adam with finite differences are good optimiser choices when considering overall final accuracy" (Section 4), are summaries of measured final energies over 372 instances, with data hosted on Zenodo. The hyperparameter and gradient sweeping described in Section 2.5 is genuine fitting: the finite-difference step size 0.4 is selected by averaging optimal step sizes on four sweeping instances, and the optimiser rankings are then obtained by running all optimisers on the full instance set. This is parameter selection, not a fitted quantity being renamed as a prediction. Self-citations to the authors' prior work appear, e.g. taking the Hamiltonian variational ansatz and initial parameters "as in [6]" and using BayesMGD from [3], but these citations supply the problem setting and candidate algorithms, not the performance ranking; the rankings are produced by the numerical runs themselves. The paper also openly acknowledges the arbitrariness in hyperparameter choice, stating "There is also much arbitrariness in what one considers a hyperparameter, and one could always promote more variables to hyperparameters and perform more in depth sweeping." The skeptical concern that tuned optimisers receive up to five hyperparameter runs while scipy optimisers receive one is a legitimate evaluation-fairness caveat, and the paper discloses this by taking "the best final energy (over different runs, e.g. for multiple hyperparameter settings)" in Fig. 6; however, this is a possible bias in the benchmark protocol, not a circular dependency in which a claimed prediction reduces to its own input. No equation in the paper defines a target quantity in terms of the fitted hyperparameters, no uniqueness theorem is imported from the authors, and no known result is merely renamed. The derivation chain is therefore self-contained with respect to circularity.

Assumptions & free parameters 3 free parameters · 3 assumptions · 0 invented entities

The central claim is an empirical ranking, so the main burden is on the choice of hyperparameters (especially the finite-difference step size) and the transfer of hyperparameters from 4 to 372 instances. No new physical entities are introduced.

free parameters (3)
  • Finite-difference step size = 0.4
    Chosen as the average of the best finite-difference step sizes found on four sweeping instances (Section 2.4). Used for all gradient-based optimizers in the main study.
  • Simultaneous perturbation step size = 0.15
    Set following [6] rather than swept; used for simultaneous perturbation gradients.
  • Per-optimizer hyperparameter sets = varies
    Best hyperparameters selected via Optuna on 4 instances; up to 5 sets per optimizer run on all instances.
assumptions (3)
  • domain assumption HV ansatz is expressive enough to approximate the ground states of the tested instances
    Tested empirically in Sec. 3.1 on exact cost functions, but not proven for all instances.
  • domain assumption Shot-noise-only cost function (without physical gate noise) is a valid proxy for near-term hardware
    Stated explicitly in Sec. 2.3: 'In this work we do not consider models incorporating physical noise in the circuit.'
  • ad hoc to paper The four instances chosen for hyperparameter sweeping are representative of all 372 instances
    The paper selects 4 instances with a note on spread, and transfers hyperparameters to all instances (Sec. 2.5).

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Cite this review

Pith. "Pith review of Benchmarking a wide range of optimisers for solving the Fermi-Hubbard model using the variational quantum eigensolver." pith.science (2026). https://pith.science/paper/JOMEX4FO

@misc{pith2026241113742,
  author       = {Pith},
  title        = {Pith review of: Benchmarking a wide range of optimisers for solving the Fermi-Hubbard model using the variational quantum eigensolver},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/JOMEX4FO}},
  note         = {Machine review of arXiv:2411.13742}
}
read the original abstract

We numerically benchmark 30 optimisers on 372 instances of the variational quantum eigensolver for solving the Fermi-Hubbard system with the Hamiltonian variational ansatz. We rank the optimisers with respect to metrics such as final energy achieved and function calls needed to get within a certain tolerance level, and find that the best performing optimisers are variants of gradient descent such as Momentum and ADAM (using finite difference), SPSA, CMAES, and BayesMGD. We also perform gradient analysis and observe that the step size for finite difference has a very significant impact. We also consider using simultaneous perturbation (inspired by SPSA) as a gradient subroutine: here finite difference can lead to a more precise estimate of the ground state but uses more calls, whereas simultaneous perturbation can converge quicker but may be less precise in the later stages. Finally, we also study the quantum natural gradient algorithm: we implement this method for 1-dimensional Fermi-Hubbard systems, and find that whilst it can reach a lower energy with fewer iterations, this improvement is typically lost when taking total function calls into account. Our method involves performing careful hyperparameter sweeping on 4 instances. We present a variety of analysis and figures, detailed optimiser notes, and discuss future directions.

Figures

Figures reproduced from arXiv: 2411.13742 by the authors.

Figure 1
Figure 1. Boxplots of optimisers tested over all 372 VQE for Fermi-Hubbard instances, ordered in terms [PITH_FULL_IMAGE:figures/full_fig_p001_1.png] view at source ↗
Figure 9
Figure 9. • The simultaneous perturbation gradient subroutine can be beneficial when seeking to minimise number of calls, whereas using finite difference is more effective for final accuracy. • We discuss the importance of hyperparameter selection, including choosing an appropriate step size for gradient subroutines (we found that for our purposes 0.4 seemed to be a good choice for finite differences). • Any improvements offe… view at source ↗
Figure 2
Figure 2. Plot of finite difference step size with absolute error (norm of difference) with the exact [PITH_FULL_IMAGE:figures/full_fig_p008_2.png] view at source ↗
Figures from the paper (28 more)
Figure 3
Figure 3. Figure 3: Gradient sweeping: boxplots of best step size found for 100 random points on four instances [PITH_FULL_IMAGE:figures/full_fig_p009_3.png]
Figure 4
Figure 4. Figure 4: We ran BFGS, L-BFGS-B, Nelder-Mead, Powell and SLSQP (from [PITH_FULL_IMAGE:figures/full_fig_p012_4.png]
Figure 5
Figure 5. Figure 5: Plots of individual runs, displaying the exact energy throughout a run (although the [PITH_FULL_IMAGE:figures/full_fig_p012_5.png]
Figure 6
Figure 6. Figure 6: This plot summarises data collected over all optimisers and all instances (372 in total). For [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: This plot is as in Fig. 6, but only displaying the best optimisers, removing the 2 [PITH_FULL_IMAGE:figures/full_fig_p013_7.png]
Figure 8
Figure 8. Figure 8: These barplots display the number of times each optimiser outperformed all the others, for [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]
Figure 9
Figure 9. Figure 9: Comparison of number of calls required to reach a certain error with the ground state energy. [PITH_FULL_IMAGE:figures/full_fig_p015_9.png]
Figure 10
Figure 10. Figure 10: For each instance, we consider on average how many calls it took each optimiser to get [PITH_FULL_IMAGE:figures/full_fig_p016_10.png]
Figure 11
Figure 11. Figure 11: Comparison of gradient-based optimisers, and of finite difference (FD) and simultaneous [PITH_FULL_IMAGE:figures/full_fig_p016_11.png]
Figure 12
Figure 12. Figure 12: Comparison of vanilla gradient descent (GD), natural gradient descent (NAT) and [PITH_FULL_IMAGE:figures/full_fig_p017_12.png]
Figure 13
Figure 13. Figure 13: Comparison of vanilla gradient descent (GD), natural gradient descent (NAT) and [PITH_FULL_IMAGE:figures/full_fig_p018_13.png]
Figure 14
Figure 14. Figure 14: Comparison of vanilla gradient descent (GD), natural gradient descent (NAT) and [PITH_FULL_IMAGE:figures/full_fig_p019_14.png]
Figure 15
Figure 15. Figure 15: This plot summarises data collected over all optimisers and all instances (372 in total). For [PITH_FULL_IMAGE:figures/full_fig_p024_15.png]
Figure 16
Figure 16. Figure 16: Comparison of the final energy difference achieved by optimisers and the best optimiser [PITH_FULL_IMAGE:figures/full_fig_p025_16.png]
Figure 17
Figure 17. Figure 17: Here we display a heatmap for some of the better performing optimisers. We see strong [PITH_FULL_IMAGE:figures/full_fig_p025_17.png]
Figure 18
Figure 18. Figure 18: Here we average each optimisers normalised (i.e. divided by the grid size [PITH_FULL_IMAGE:figures/full_fig_p026_18.png]
Figure 19
Figure 19. Figure 19: Hill climber hyperparameters. C.2 Coordinate Descent - Overview: The algorithm proceeds by minimising each parameter in turn, and is discussed in detail in [6]. For each parameter (i.e. keeping all others fixed), a trigonometric interpolating polynomial is constructed…
Figure 20
Figure 20. Figure 20: Coordinate descent hyperparameters. C.3 BayesMGD - Overview: Bayes Model Gradient Descent (BayesMGD) was introduced in [3] as an extension to Model Gradient Descent [18]. - Implementation: in house. - Evaluations per iteration: η 2 (ν + 1)(ν + 2) - Black box: Yes, but…
Figure 21
Figure 21. Figure 21: BayesMGD hyperparameters. C.4 SPSA - Overview: The simultaneous perturbation stochastic approximation (SPSA) algorithm follows a standard gradient descent structure, but uses a noisy approximation to the gradient using only two evaluations. In particular, for each par…
Figure 22
Figure 22. Figure 22: SPSA hyperparameters. C.5 Gradient Descent - Overview: Basic (or vanilla) gradient descent simply takes a step in the direction of negative gradient (i.e. downhill) at each iteration. A priori, the algorithm assumes access to some gradient function ∇ f , and takes as …
Figure 23
Figure 23. Figure 23: Gradient descent hyperparameters. C.6 Momentum - Overview: One can extend vanilla gradient descent by including a velocity type term. vt+1 = γvt + η∇ f(θt) (50) θt+1 = θt − vt (51) A variant of this is known as nesterov, in which we instead evaluate the gradient at th…
Figure 24
Figure 24. Figure 24: Momentum hyperparameters. C.7 AdaDelta - Overview: AdaDelta [52] is an extension to Momentum (more specifically, to another method known as AdaGrad [53]). Some of the motivations for this optimiser include removing the need to manually set a learning rate, using previ…
Figure 25
Figure 25. Figure 25: AdaDelta hyperparameters. C.8 Adam - Overview: Adaptive Moment Estimation (Adam) [54] has become a popular method in machine learning, storing exponentially decaying averages of the first and second moments of the gradient, and then uses a similar update step to that …
Figure 26
Figure 26. Figure 26: Adam hyperparameters. C.9 λ + µ - Overview: In a basic evolutionary strategy, at each iteration (generation), µ ‘parents’ are selected to produce λ ‘offspring’. Two functions must be specified: how to combine two solutions to produce a child solution (crossover, matin…
Figure 27
Figure 27. Figure 27: λ + µ hyperparameters. C.10 Particle Swarm Optimisation - Overview: Particle Swarm Optimisation (PSO) is an evolutionary algorithm consisting of a set of candidate solutions (particles) exploring the search space. At each iteration, the location and velocity of each p…
Figure 28
Figure 28. Figure 28: PSO hyperparameters. C.11 CMAES - Overview: The Covariance Matrix Adaption Evolution Strategy (CMAES) is a black-box nu￾merical optimiser. At each generation, new candidate solutions are sampled from a multivari￾ate normal distribution subject to a covariance matrix (…
Figure 29
Figure 29. Figure 29: CMAES hyperparameters. C.12 Nelder-Mead - Implementation: scipy [34]. - Evaluations per iteration: ν + 1 (in the worst case). - Black box: Yes. - gradient-based: No. - References: [34, 43, 44]. - Pseudocode: [10]. - Hyperparameters: 41 [PITH_FULL_IMAGE:figures/full_f…
Figure 30
Figure 30. Figure 30: Nelder-Mead hyperparameters. C.13 Other optimisers COBYLA - Implementation: scipy [34]. - Black box: Yes. - gradient-based: No. - References: [45]. BFGS - Implementation: scipy [34]. - Black box: Yes. - gradient-based: Yes. - References: [8, 34]. L-BFGS-B - Implementa…

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Pith tools

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