Pith. sign in

REVIEW 3 major objections 4 minor 73 references

Looking elsewhere: improving variational Monte Carlo gradients by importance sampling

T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash

Challenge this review Re-run · record.json Download PDF Read on arXiv ↗

Pith's one-line read Sampling from $|\psi_\theta|^\alpha$ with $\alpha$ tuned each step to maximize gradient signal-to-noise cuts vanilla variational Monte Carlo cost up to 100× on peaked molecular wavefunctions, with no new hyperparameters.

desk verdict A practical adaptive importance-sampling scheme for VMC with sound variance math and honest benchmarks, but the 100x headline and the bare-gradient-vs-SR objective gap need to be handled in revision. read the letter →

arxiv 2507.05352 v1 pith:AGQQ2HEF submitted 2025-07-07 quant-ph cond-mat.dis-nncond-mat.stat-mechcond-mat.str-el

Antoine Misery , Luca Gravina , Alessandro Santini , Filippo Vicentini This is my paper
classification quant-phcond-mat.dis-nncond-mat.stat-mechcond-mat.str-el
keywords variationalMonteCarloimportancesamplingneuralquantumstatesgradientsignal-to-noiseratiooverdispersedchemistrystochasticreconfigurationprojecteddynamics
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

Variational Monte Carlo training of neural quantum states usually estimates gradients from samples of the Born distribution $|\psi_\theta(x)|^2$; for sharply peaked wavefunctions, typical of quantum chemistry, that estimator has such high variance that training slows, stalls, or fails. The paper argues that sampling from the flattened family $q_\alpha(x) \propto |\psi_\theta(x)|^\alpha$ and adaptively tuning $\alpha$ to maximize the average gradient signal-to-noise ratio systematically reduces that variance. The tuning rule costs almost nothing, because it reuses the weights and local gradients already computed in a natural-gradient step, and it introduces no additional hyperparameters. Across frustrated spin systems, molecular ground states, and projected quantum dynamics, the adaptive sampler beats Born sampling and fixed overdispersion, and for peaked molecular wavefunctions it cuts the computational cost of vanilla VMC by up to a factor of 100.

What carries the argument

The central object is the one-parameter overdispersed sampling family $q_\alpha(x) = |\psi_\theta(x)|^\alpha / Z_\alpha$, $0 \le \alpha \le 2$, which interpolates between uniform sampling ($\alpha = 0$) and Born sampling ($\alpha = 2$). The objective that carries the argument is the average gradient signal-to-noise ratio $\mathcal{L}_{\mathrm{IS}}(q) = \frac{1}{N_p}\sum_i \mathrm{SNR}_q(f_i^q)$, built from the self-normalized importance-sampling estimator $f_i^q(x) = W(x) f_i(x)$ with weights $W(x) = \frac{|\psi_\theta(x)|^2}{q(x)}\frac{Z_q}{Z_\theta}$ and $\mathrm{SNR}_q(f_i^q) = |\mathbb{E}_q[f_i^q]|\big/\sqrt{\mathbb{V}_q[f_i^q]}$. The load-bearing identity is the variance formula $\mathbb{V}_q[f_i^q] = \mathbb{E}_q\big[W^2 |f_i - F_i|^2\big]$, derived by a delta-method expansion, which converts the $\alpha$-derivative of the variance into the computable covariance $\partial_\alpha \mathbb{V}_{q_\alpha}[f_i^{q_\alpha}] = -\mathrm{Cov}_{q_\alpha}(\partial_\alpha \log q_\alpha,\, g_i^2)$ with $g_i = W|f_i - F_i|$. Equations (12)–(14) assemble this into the adaptive rule $\alpha' = \alpha + \eta\,\partial_\alpha \mathcal{L}_{\mathrm{IS}}$ with a fixed learning rate $\eta = 0.1$ and per-step clipping, all evaluated on the same Monte Carlo batch that produces the gradient and the quantum geometric tensor.

What would settle it

On a small system whose Hilbert space can be summed exactly — the paper itself treats N$_2$ in the STO-3G basis this way — compute, along one training run, both the $\alpha$ the adaptive rule chooses and the $\alpha$ that exactly maximizes the signal-to-noise ratio of the full stochastic-reconfiguration update, metric tensor included. If the exact maximizer systematically disagrees with the adaptive choice, or if an optimization driven by it reaches lower energy with fewer samples, the paper's bare-gradient objective is the wrong proxy and its reported gains would not transfer to natural-gradient regimes where metric-tensor noise dominates.

Watch

Extended reading notes

Core claim

The paper's central claim is that the Monte Carlo variance that limits variational Monte Carlo training is a property of the estimator under the chosen sampling distribution, and that the standard Born choice $p(x) = |\psi_\theta(x)|^2$ is typically suboptimal. Sampling instead from the overdispersed family $q_\alpha(x) \propto |\psi_\theta(x)|^\alpha$ with $0 \le \alpha \le 2$ flattens the tails of the distribution, and tuning $\alpha$ adaptively to maximize the component-averaged gradient signal-to-noise ratio $\mathcal{L}_{\mathrm{IS}}$ reduces gradient-estimator variance — sometimes by orders of magnitude — at no additional computational cost and with no per-problem hyperparameter search. The tuning identity is that the $\alpha$-derivative of the estimator variance is a covariance that can be estimated from the same samples used for the gradient and the metric tensor, so the update $\alpha' = \alpha + \eta\,\partial_\alpha \mathcal{L}_{\mathrm{IS}}$ with fixed $\eta = 0.1$ costs essentially nothing inside stochastic reconfiguration. The authors validate the approach a posteriori across frustrated spin models, molecular ground states, and projected quantum dynamics: the adaptive sampler beats Born sampling and all fixed overdispersion choices, and on peaked quantum chemistry wavefunctions it reduces the computational cost of vanilla VMC by up to a factor of 100.

Load-bearing premise

The load-bearing premise — flagged by the paper at the end of Section II.E — is that maximizing the average signal-to-noise ratio of the bare gradient makes VMC training faster and more stable; if that number is the wrong proxy for the natural-gradient update actually used, the tuned sampler optimizes an irrelevant objective.

Editorial extensions

If this is right

  • The method can be dropped into existing VMC and stochastic-reconfiguration code with no change in per-step complexity, since the weights, local gradients, and variance estimators needed for the $\alpha$-update are already computed in a natural-gradient step.
  • For Li$_2$O in the STO-3G basis, adaptive overdispersion reaches the reference accuracy obtained with $2^{14}$ selected-configuration samples using only $2^{11}$ samples, while Born sampling misses by about two orders of magnitude at the same budget.
  • At the maximally frustrated point $J_2/J_1 = 0.5$ of the square $J_1$-$J_2$ model, adaptive sampling improves the relative energy error by roughly half an order of magnitude at fixed sample count and consistently outperforms every fixed choice of $\alpha$.
  • In projected time-dependent VMC, infidelity minimization reproduces the exact-summation result with $2^{13}$ samples, whereas Born sampling needs at least $2^{17}$ samples — more than the $2^{16}$ configurations in the Hilbert space.
  • The exact signal-to-noise analysis implies that near convergence on peaked molecular states, Born sampling needs roughly 1000× more samples than the optimally overdispersed distribution merely to match the gradient signal-to-noise ratio.

Reading between the lines

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

  • The paper validates its bare-gradient objective only after the fact; a sharper test would tune $\alpha$ against the signal-to-noise ratio of the full stochastic-reconfiguration update, metric tensor included, and compare the resulting optimizers — if the preferred $\alpha$ differs, the reported speedups may not carry over to natural-gradient regimes where metric noise dominates.
  • The appendix's bias analysis shows that the self-normalized estimator's shrinkage grows as $\alpha \to 0$ and is negligible only at the sample sizes used; at very small sample budgets or with $\alpha$ pushed toward 0, a bias-corrected or defensive-mixture estimator would be needed to keep the method safe.
  • Because $\mathcal{L}_{\mathrm{IS}}$ is exactly computable on small systems, one could track the gap between the adaptively chosen $\alpha$ and the exact maximizer of $\mathcal{L}_{\mathrm{IS}}$ during training, separating the cost of the gradient-ascent tuning rule from any limitation of the overdispersed family itself.
  • A natural extension, suggested by the SNR stability criterion the paper quotes, is to adapt the sample size together with $\alpha$ so the gradient signal-to-noise ratio stays above a fixed threshold; that would turn the reported 100× cost reduction into a scheduling choice rather than a post-hoc comparison.
Share X Bluesky LinkedIn Reddit HN

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 manuscript proposes a variance-reduction scheme for variational Monte Carlo (VMC) gradient estimation in which configurations are sampled from the overdispersed family q_alpha(x) ∝ |ψ_θ(x)|^α, with α tuned adaptively to maximize the average signal-to-noise ratio L_IS of the bare gradient estimator. The authors derive the self-normalized importance sampling (SNIS) variance and bias, give a gradient-based update rule for α, and benchmark the method on frustrated spin systems, the N2 and Li2O molecules, and infidelity minimization in projected dynamics. They report large reductions in required sample counts and claim up to a factor of 100x reduction in computational cost for peaked quantum-chemistry wavefunctions.

Significance. If the central claim is validated, the method is a simple, low-cost, and broadly applicable improvement to NQS training. The manuscript has concrete strengths: the SNIS variance and bias derivations in Appendix A.3 are careful and correct; the exact full-summation SNR analysis in Fig. 4 directly supports the claim that α can be tuned to track the gradient SNR; and the authors state that code and data will be released. The significance is currently tempered by two issues: the adaptive rule optimizes the SNR of the bare gradient while the headline benchmarks use stochastic reconfiguration, and the 'up to 100x' abstract claim is not directly supported by the presented end-to-end benchmarks.

major comments (3)
  1. [Section II.E and Eqs. (15)–(16)] The paper explicitly acknowledges in Section II.E that it 'focus[es] exclusively on maximizing the SNR of the bare gradient when comparing different sampling distributions,' while all headline benchmarks use stochastic reconfiguration, whose update (Eq. (15)) involves the metric tensor S estimated from the same q_alpha samples. Since changing α changes the self-normalized weights W and hence the noise and conditioning of S, the SNR of the preconditioned update (S + λ)^{-1}g is not a function of the bare-gradient SNR alone. The a-posteriori validation in Figs. 2, 3, and 5 does not isolate this assumption because α is always adapted by the same L_IS objective; no experiment tests whether a different α that maximizes the SNR of the SR update would converge faster or more stably. I request a controlled test of this load-bearing assumption, for example a fixed-state computation of the SNR of the preconditioned update components as a function of α, compared against the L_IS-optimal α, or a benchmark comparing convergence with α chosen by each objective.
  2. [Abstract and Section III.C] The abstract's claim that the approach 'can reduce the computational cost of vanilla VMC considerably, up to a factor of 100x' is not directly supported by the end-to-end benchmarks. In the Li2O results (Section III.B and Fig. 3), the adaptive method with N_s = 2^11 samples matches an FSSC calculation with 2^14 samples, an 8x sample-count reduction, and the infidelity benchmark (Fig. 5) shows a 16x reduction (2^13 vs 2^17 samples). The N2 exact-SNR analysis in Section III.C is about the SNR of gradient estimators, not end-to-end cost; a two-order-of-magnitude SNR gap corresponds to a much larger sample-count factor, and no wall-clock or sample-to-target-error comparison is presented there. Please either moderate the abstract claim to match the demonstrated gains or provide a direct cost comparison, such as samples-to-target-error curves for adaptive vs Born sampling with matched total computational cost.
  3. [Equation (13) and Appendix B.4] Equation (13) as printed gives ∂_α SNR = ∂_α log V / (2 SNR), which differs from the correct expression in Eq. (B4), namely (1/2) ∂_α V |E| / V^{3/2}, by a factor of SNR^2. The sign is unchanged, so this is likely a typographical error, but it is load-bearing: a reader implementing the adaptive rule from the main text would obtain an incorrectly scaled α update, and since the missing factor is SNR-dependent, this is not equivalent to a simple rescaling of η. Please correct Eq. (13) or clearly cross-reference Eq. (B4) as the implementation formula.
minor comments (4)
  1. [Abstract and Section II.D] The abstract lists 'not introduce additional hyperparameters' as a design goal, but Section II.D introduces η = 0.1 and a per-step clip δ = 0.01. These are fixed rather than per-problem tuned, so the claim should be rephrased, e.g., 'no per-problem hyperparameter tuning required.'
  2. [Section III.C] The text states that there is 'at most a two-orders-of-magnitude difference between the SNR' and then that the required number of samples 'must be increased by at least a factor of 1000.' Since SNR scales as the square root of the sample count, a two-order-of-magnitude SNR gap corresponds to a factor of about 10^4 in sample count, not 10^3. Please clarify the intended relationship.
  3. [Section III.C] There is a typo in 'A pratical stability criterion' — should be 'practical.'
  4. [Figure 2 caption] The caption says 'In both cases, the adaptive strategy (black) outperforms the fixed overdispersion coefficient,' but the figure contains three panels (J2/J1 = 0, 0.5, 1.0). Please rephrase to refer to all three cases.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: adaptive α maximizes gradient SNR, while the reported gains are measured on independent energy and infidelity objectives.

full rationale

Walking the derivation chain: the paper defines the family q_α ∝ |ψ|^α (Eq. 11), the average gradient-SNR objective L_IS (Eq. 8), and the adaptive rule α′ = α + η ∂_α L_IS (Eq. 12), with ∂_α L_IS derived in Eqs. (13)-(14) and Appendix A.3. L_IS is a variance-reduction objective independently motivated by Hesterberg's optimal importance-sampling distribution (Eq. 10) and the delta-method variance of self-normalized estimators (Appendix A.3). The headline results, however, are not stated in terms of L_IS: they are relative energy errors (Figs. 2-3), exact full-basis SNR comparisons (Fig. 4), and final infidelities (Fig. 5). The α that maximizes SNR is therefore not constructed to minimize the quantity in which improvement is claimed, so the improvements are not forced by the definition of the update. Section II.E explicitly labels the bare-gradient-SNR proxy as an assumption to be 'validated a posteriori by the performance of our approach'; the validation is empirical rather than definitional. Self-citations to Refs. [23,24] supply background and an SNR heuristic, but neither is a uniqueness theorem, and the current benchmarks plus the independent Malyshev et al. result provide external backing. The derivative in Eq. (13) appears to contain a typo inconsistent with Eq. (B4), but a wrong derivative is a correctness issue, not circularity. No equation or fitted parameter is renamed as a prediction; the derivation is self-contained.

Assumptions & free parameters 3 free parameters · 4 assumptions · 0 invented entities

The central claim rests on the SNR proxy assumption (Section II.E), on the delta-method variance formula (Appendix A.3), and on the practical validity of sampling from a slowly moving q_alpha. Three hand-set constants (eta, clip, and the adaptive alpha itself) are added despite the abstract's 'no additional hyperparameters' claim.

free parameters (3)
  • alpha (overdispersion exponent) = adaptive, approximately 0.8 to 1.8 depending on system and stage
    Tuned at each step by gradient ascent on L_IS (Eq. 12); the whole method rests on this adaptive parameter.
  • eta (learning rate for alpha) = 0.1
    Fixed by hand in Section II.D; claimed robust across all tested cases.
  • delta (per-step alpha clip) = 0.01
    Hand-set maximum increment of alpha per step in Section II.D to preserve MCMC thermalization.
assumptions (4)
  • domain assumption The average gradient SNR L_IS is the correct objective to optimize for VMC convergence
    Stated in Section II.E and validated only a posteriori by the benchmarks.
  • standard math First-order delta method gives an accurate variance expression for the self-normalized importance sampling estimator
    Used in Appendix A.3.b to derive Eq. (9) and Eq. (A4); higher-order terms are neglected.
  • domain assumption Monte Carlo samples are effectively drawn from q_alpha despite alpha changing by up to 0.01 per step
    Section II.D clips updates to preserve thermalization, but no convergence guarantee for a moving target distribution is given; with acceptance rates as low as 1e-4 (Fig. 3) this is fragile.
  • domain assumption The self-normalized importance sampling bias is negligible at the sample sizes used
    Appendix A.3.c shows the shrinkage factor rho0 is on the order of 10 for N2, small relative to 1/n for n in the 10^3 to 10^4 range, but this is checked only for one state.

how reviews work

0 comments
Cite this review

Pith. "Pith review of Looking elsewhere: improving variational Monte Carlo gradients by importance sampling." pith.science (2026). https://pith.science/paper/AGQQ2HEF

@misc{pith2026250705352,
  author       = {Pith},
  title        = {Pith review of: Looking elsewhere: improving variational Monte Carlo gradients by importance sampling},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/AGQQ2HEF}},
  note         = {Machine review of arXiv:2507.05352}
}
read the original abstract

Neural-network quantum states (NQS) offer a powerful and expressive ansatz for representing quantum many-body wave functions. However, their training via Variational Monte Carlo (VMC) methods remains challenging. It is well known that some scenarios - such as sharply peaked wave functions emerging in quantum chemistry - lead to high-variance gradient estimators hindering the effectiveness of variational optimizations. In this work we investigate a systematic strategy to tackle those sampling issues by means of adaptively tuned importance sampling. Our approach is explicitly designed to (i) target the gradient estimator instead of the loss function, (ii) not introduce additional hyperparameters and (iii) be computationally inexpensive. We benchmarked our approach across the ground-state search of a wide variety of hamiltonians, including frustrated spin systems and ab-initio quantum chemistry. We also show systematic improvements on the infidelity minimization in the context of neural projected quantum dynamics. Overall, our approach can reduce the computational cost of vanilla VMC considerably, up to a factor of 100x when targeting highly peaked quantum chemistry wavefunctions.

Figures

Figures reproduced from arXiv: 2507.05352 by the authors.

Figure 1
Figure 1. FIG. 1. Component-wise averaged signal-to-noise ratio [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. FIG. 2. Scaling of relative error with sample size for a square [PITH_FULL_IMAGE:figures/full_fig_p005_2.png] view at source ↗
Figure 3
Figure 3. FIG. 3. ( [PITH_FULL_IMAGE:figures/full_fig_p006_3.png] view at source ↗
Figures from the paper (5 more)
Figure 4
Figure 4. Figure 4: FIG. 4. Mean gradient Signal-to-Noise ratio, for several importance sampling distribution and systems. ( [PITH_FULL_IMAGE:figures/full_fig_p007_4.png]
Figure 5
Figure 5. Figure 5: FIG. 5. Effect of varying the overdispersion parameter [PITH_FULL_IMAGE:figures/full_fig_p008_5.png]
Figure 6
Figure 6. Figure 6: FIG. 6. Effective sample size and SNR curves over [PITH_FULL_IMAGE:figures/full_fig_p013_6.png]
Figure 7
Figure 7. Figure 7: FIG. 7. Comparison of several objective for the adaptive tun [PITH_FULL_IMAGE:figures/full_fig_p014_7.png]
Figure 8
Figure 8. Figure 8: FIG. 8. Comparison of SPRING with [PITH_FULL_IMAGE:figures/full_fig_p014_8.png]

Discussion (0). Sign in to comment.

Reference graph

Works this paper leans on

73 extracted references · 55 canonical work pages

  1. [1]

    Finally, it is important to mention that because of er- godicity, VMC can still perform decently even though the SNR of the gradient is low

    This could provide a criterion for adaptive sample size; Indeed, during the VMC optimization one can dynami- cally adjust the sample size to maintain √NsLIS(q) ≳ 1. Finally, it is important to mention that because of er- godicity, VMC can still perform decently even though the SNR of the gradient is low. However, a higher SNR sys- tematically yields great...

    work page 2024

  2. [2]

    Sharir, Y

    O. Sharir, Y. Levine, N. Wies, G. Carleo, and A. Shashua, Deep Autoregressive Models for the Efficient Variational Simulation of Many-Body Quantum Systems, Physical Review Letters 124, 020503 (2020), publisher: American Physical Society

    work page 2020

  3. [3]

    Carleo and M

    G. Carleo and M. Troyer, Solving the quantum many- body problem with artificial neural networks, Science 355, 602 (2017), publisher: American Association for the Advancement of Science

    work page 2017

  4. [4]

    M. Reh, M. Schmitt, and M. G¨ arttner, Optimizing design choices for neural quantum states, Physical Review B 107, 10.1103/physrevb.107.195115 (2023)

    work page doi:10.1103/physrevb.107.195115 2023

  5. [5]

    K. Choo, G. Carleo, N. Regnault, and T. Neupert, Symmetries and Many-Body Excitations with Neural- Network Quantum States, Physical Review Letters 121, 167204 (2018), publisher: American Physical Society

    work page 2018

  6. [6]

    Luo and B

    D. Luo and B. K. Clark, Backflow Transformations via Neural Networks for Quantum Many-Body Wave Func- tions, Physical Review Letters 122, 226401 (2019), pub- lisher: American Physical Society

    work page 2019

  7. [7]

    R. P. Nutakki, A. Shokry, and F. Vicentini, Design prin- ciples of deep translationally-symmetric neural quantum states for frustrated magnets (2025)

    work page 2025

  8. [8]

    Robledo Moreno, G

    J. Robledo Moreno, G. Carleo, A. Georges, and J. Stokes, Fermionic wave functions from neural-network con- strained hidden states, Proceedings of the National Academy of Sciences 119, e2122059119 (2022), publisher: Proceedings of the National Academy of Sciences

    work page 2022

Show all 73 references
  1. [9]

    K. Choo, A. Mezzacapo, and G. Carleo, Fermionic neural-network states for ab-initio electronic structure, Nature Communications 11, 2368 (2020), 1909.12852

    2020 arXiv

  2. [11]

    D. Luo, G. Carleo, B. K. Clark, and J. Stokes, Gauge equivariant neural networks for quantum lattice gauge theories, Physical Review Letters 127, 10.1103/phys- revlett.127.276402 (2021)

    2021 doi

  3. [12]

    Torlai and R

    G. Torlai and R. G. Melko, Latent space purification via neural density operators, Physical Review Letters 120, 10.1103/physrevlett.120.240503 (2018)

    2018 doi

  4. [13]

    A. Apte, C. C´ ordova, T.-C. Huang, and A. Ashmore, Deep learning lattice gauge theories, Physical Review B 110, 10.1103/physrevb.110.165133 (2024)

    2024 doi

  5. [14]

    Vicentini, A

    F. Vicentini, A. Biella, N. Regnault, and C. Ciuti, Variational neural-network ansatz for steady states in open quantum systems, Physical Review Letters 122, 10.1103/physrevlett.122.250503 (2019)

    2019 doi

  6. [15]

    Vicentini, R

    F. Vicentini, R. Rossi, and G. Carleo, Positive-definite parametrization of mixed quantum states with deep neu- ral networks (2022), 2206.13488

    2022 arXiv

  7. [16]

    Eeltink, F

    D. Eeltink, F. Vicentini, and V. Savona, Variational dy- namics of open quantum systems in phase space (2023)

    2023

  8. [17]

    D. Luo, Z. Chen, J. Carrasquilla, and B. K. Clark, Au- toregressive neural network for simulating open quantum systems via a probabilistic formulation, Physical Review Letters 128, 10.1103/physrevlett.128.090501 (2022)

    2022 doi

  9. [18]

    Chen and M

    A. Chen and M. Heyl, Empowering deep neural quantum states through efficient optimization, Nature Physics 20, 1476 (2024), publisher: Nature Publishing Group

    2024

  10. [19]

    M. Reh, M. Schmitt, and M. G¨ arttner, Time-dependent variational principle for open quantum systems with ar- tificial neural networks, Phys. Rev. Lett. 127, 230501 (2021)

    2021

  11. [20]

    Sorella, Green Function Monte Carlo with Stochas- tic Reconfiguration, Physical Review Letters 80, 4558 (1998), publisher: American Physical Society

    S. Sorella, Green Function Monte Carlo with Stochas- tic Reconfiguration, Physical Review Letters 80, 4558 (1998), publisher: American Physical Society

    1998

  12. [21]

    S. Dash, L. Gravina, F. Vicentini, M. Ferrero, and A. Georges, Efficiency of neural quantum states in light of the quantum geometric tensor, Communications Physics 8, 10.1038/s42005-025-02005-4 (2025)

    2025 doi

  13. [22]

    Martens, New insights and perspectives on the natural gradient method, Journal of Machine Learning Research 21, 1 (2020)

    J. Martens, New insights and perspectives on the natural gradient method, Journal of Machine Learning Research 21, 1 (2020)

    2020

  14. [23]

    Amari, Natural gradient works efficiently in learning, Neural Computation 10, 251 (1998)

    S.-i. Amari, Natural gradient works efficiently in learning, Neural Computation 10, 251 (1998)

    1998

  15. [24]

    Gravina, V

    L. Gravina, V. Savona, and F. Vicentini, Neural Pro- jected Quantum Dynamics: a systematic study (2024), 2410.10720

    2024 arXiv

  16. [25]

    Sinibaldi, C

    A. Sinibaldi, C. Giuliani, G. Carleo, and F. Vicentini, Unbiasing time-dependent Variational Monte Carlo by projected quantum evolution, Quantum 7, 1131 (2023), 2305.14294

    2023 arXiv

  17. [26]

    Park and M

    C.-Y. Park and M. J. Kastoryano, Expressive power of complex-valued restricted Boltzmann machines for solv- ing non-stoquastic Hamiltonians (2022), 2012.08889

    2022 arXiv

  18. [27]

    Chen and M

    A. Chen and M. Heyl, Efficient optimization of deep neural quantum states toward machine precision (2023), 2302.01941v2

    2023 arXiv

  19. [28]

    Malyshev, M

    A. Malyshev, M. Schmitt, and A. I. Lvovsky, Neural Quantum States and Peaked Molecular Wave Functions: Curse or Blessing? (2024), 2408.07625

    2024 arXiv

  20. [29]

    Li, J.-C

    X. Li, J.-C. Huang, G.-Z. Zhang, H.-E. Li, C.-s. Cao, D. Lv, and H.-S. Hu, A Non-stochastic Optimization Algorithm for Neural-network Quantum States, Journal of Chemical Theory and Computation 19, 8156 (2023), 2305.12776

    2023 arXiv

  21. [30]

    Li, J.-C

    X. Li, J.-C. Huang, G.-Z. Zhang, H.-E. Li, Z.-P. Shen, C. Zhao, J. Li, and H.-S. Hu, Improved optimization for the neural-network quantum states and tests on the chromium dimer, The Journal of Chemical Physics 160, 234102 (2024)

    2024

  22. [31]

    Liu and B

    A.-J. Liu and B. K. Clark, Efficient optimization of neural network backflow for ab-initio quantum chemistry (2025), 2502.18843

    2025 arXiv

  23. [32]

    Hibat-Allah, M

    M. Hibat-Allah, M. Ganahl, L. E. Hayward, R. G. Melko, and J. Carrasquilla, Recurrent neural network wave func- tions, Physical Review Research 2, 023358 (2020), pub- lisher: American Physical Society

    2020

  24. [33]

    X. Cao, Z. Zhong, and Y. Lu, Vision Transformer Neural Quantum States for Impurity Models (2024), 2408.13050

    2024 arXiv

  25. [34]

    T. Zhao, J. Stokes, and S. Veerapaneni, Scalable neu- ral quantum states architecture for quantum chemistry, Machine Learning: Science and Technology 4, 025034 (2023), publisher: IOP Publishing

    2023

  26. [35]

    Malyshev, J

    A. Malyshev, J. M. Arrazola, and A. I. Lvovsky, Autore- gressive Neural Quantum States with Quantum Number Symmetries (2023), 2310.04166

    2023 arXiv

  27. [36]

    Bravyi, G

    S. Bravyi, G. Carleo, D. Gosset, and Y. Liu, A rapidly mixing Markov chain from any gapped quantum many- body system, Quantum 7, 1173 (2023), 2207.07044

    2023 arXiv

  28. [37]

    A. A. Bagrov, A. A. Iliasov, and T. Westerhout, Kinetic samplers for neural quantum states, Physical Review B 10 104, 104407 (2021), 2011.02986

    2021 arXiv

  29. [38]

    Katharopoulos and F

    A. Katharopoulos and F. Fleuret, Not All Samples Are Created Equal: Deep Learning with Importance Sam- pling, in Proceedings of the 35th International Confer- ence on Machine Learning (PMLR, 2018) pp. 2525–2534, iSSN: 2640-3498

    2018

  30. [39]

    Alain, A

    G. Alain, A. Lamb, C. Sankar, A. Courville, and Y. Ben- gio, Variance Reduction in SGD by Distributed Impor- tance Sampling (2016), 1511.06481

    2016 arXiv

  31. [40]

    F. J. R. Ruiz, M. K. Titsias, and D. M. Blei, Overdispersed Black-Box Variational Inference (2016), 1603.01140

    2016 arXiv

  32. [41]

    R. Liu, T. Wu, and B. Mozafari, Adam with Bandit Sam- pling for Deep Learning, ArXiv (2020)

    2020

  33. [42]

    Grenioux, E

    L. Grenioux, E. Moulines, and M. Gabri´ e, Balanced Training of Energy-Based Models with Adaptive Flow Sampling (2024), 2306.00684

    2024 arXiv

  34. [43]

    X. Li, C. Li, J. Chi, and J. Ouyang, Variance Reduc- tion in Black-box Variational Inference by Adaptive Im- portance Sampling, in Proceedings of the Twenty-Seventh International Joint Conference on Artificial Intelligence (International Joint Conferences on Artificial Intelli...

    2018

  35. [44]

    Schmitt and M

    M. Schmitt and M. Heyl, Quantum many-body dynamics in two dimensions with artificial neural networks (2020), 1912.08828

    2020 arXiv

  36. [45]

    K. Inui, Y. Kato, and Y. Motome, Determinant-free fermionic wave function using feed-forward neural net- works, Physical Review Research 3, 10.1103/physrevre- search.3.043126 (2021)

    2021 doi

  37. [46]

    Novak, J

    R. Novak, J. Sohl-Dickstein, and S. S. Schoenholz, Fast finite width neural tangent kernel (2022)

    2022

  38. [47]

    Rende, L

    R. Rende, L. L. Viteritti, L. Bardone, F. Becca, and S. Goldt, A simple linear algebra identity to optimize large-scale neural network quantum states, Communica- tions Physics 7, 1 (2024), publisher: Nature Publishing Group

    2024

  39. [48]

    We do not include a mixture of the exact optimal distri- butions qi opt as estimating each Fi at every MCMC step would be computationally prohibitive and largely redun- dant

  40. [49]

    T. Hesterberg, Weighted Average Importance Sampling and Defensive Mixture Distributions, Technometrics 37, 185 (1995), publisher: [Taylor & Francis, Ltd., American Statistical Association, American Society for Quality]

    1995

  41. [50]

    Bukov, M

    M. Bukov, M. Schmitt, and M. Dupont, Learning the ground state of a non-stoquastic quantum Hamiltonian in a rugged neural network landscape (2021), 2011.11214

    2021 arXiv

  42. [51]

    Marshall, Antiferromagnetism, Proceedings of the Royal Society of London

    W. Marshall, Antiferromagnetism, Proceedings of the Royal Society of London. Series A, Mathematical and Physical Sciences 232, 48 (1955), publisher: The Royal Society

    1955

  43. [52]

    Becca and S

    F. Becca and S. Sorella, Quantum Monte Carlo Ap- proaches for Correlated Systems (Cambridge University Press, 2017)

    2017

  44. [53]

    Nocedal and S

    J. Nocedal and S. J. Wright, Numerical Optimization , Springer Series in Operations Research and Financial En- gineering, Vol. 2 (Springer New York, 2006)

    2006

  45. [54]

    W. Kool, H. v. Hoof, and M. Welling, Stochastic Beams and Where to Find Them: The Gumbel-Top-k Trick for Sampling Sequences Without Replacement (2019), 1903.06059

    2019 arXiv

  46. [55]

    We use a neural network backflow with pa- rameters identical to the literature [29, 56]

    database, using the STO-3G basis and canonical Hartree-Fock (HF) orbitals; in that setting, dim( H) = 4.14 × 107. We use a neural network backflow with pa- rameters identical to the literature [29, 56]. While it has been identified as a particularly challenging molecule for wh...

  47. [56]

    L. L. Viteritti, R. Rende, and F. Becca, Transformer Vari- ational Wave Functions for Frustrated Quantum Spin Systems, Physical Review Letters 130, 236401 (2023), publisher: American Physical Society

    2023

  48. [57]

    PubChem Substance and Compound databases | Nucleic Acids Research | Oxford Academic

  49. [58]

    Liu and B

    A.-J. Liu and B. K. Clark, Neural network backflow for ab initio quantum chemistry, Physical Review B 110, 115137 (2024), publisher: American Physical Society

    2024

  50. [59]

    Goldshlager, N

    G. Goldshlager, N. Abrahamsen, and L. Lin, A Kaczmarz-inspired approach to accelerate the optimiza- tion of neural network wavefunctions (2024), 2401.10190

    2024 arXiv

  51. [60]

    Nys, Data accompanying ”ab-initio variational wave functions for the time-dependent many-electron schr¨ odinger equation” (2024)

    J. Nys, Data accompanying ”ab-initio variational wave functions for the time-dependent many-electron schr¨ odinger equation” (2024)

    2024

  52. [61]

    Vicentini, D

    F. Vicentini, D. Hofmann, A. Szab´ o, D. Wu, C. Roth, C. Giuliani, G. Pescia, J. Nys, V. Vargas-Calderon, N. Astrakhantsev, and G. Carleo, NetKet 3: Machine Learning Toolbox for Many-Body Quantum Systems, Sci- Post Physics Codebases , 7 (2022), 2112.10526

    2022 arXiv

  53. [62]

    Carleo, K

    G. Carleo, K. Choo, D. Hofmann, J. E. T. Smith, T. Westerhout, F. Alet, E. J. Davis, S. Efthymiou, I. Glasser, S.-H. Lin, M. Mauri, G. Mazzola, C. B. Mendl, E. van Nieuwenburg, O. O’Reilly, H. Th´ eveniaut, G. Tor- lai, F. Vicentini, and A. Wietek, NetKet: A machine learning t...

    2019

  54. [63]

    H¨ afner and F

    D. H¨ afner and F. Vicentini, mpi4jax: Zero-copy MPI communication of JAX arrays, Journal of Open Source Software 6, 3419 (2021)

    2021

  55. [64]

    Bradbury, R

    J. Bradbury, R. Frostig, P. Hawkins, M. J. Johnson, C. Leary, D. Maclaurin, G. Necula, A. Paszke, J. Van- derPlas, S. Wanderman-Milne, and Q. Zhang, JAX: com- posable transformations of Python+NumPy programs (2018)

    2018

  56. [65]

    J. Heek, A. Levskaya, A. Oliver, M. Ritter, B. Ronde- pierre, A. Steiner, and M. van Zee, Flax: A neural net- work library and ecosystem for JAX (2024)

    2024

  57. [66]

    G. W. Oehlert, A note on the delta method, The Amer- ican Statistician 46, 27 (1992). Appendix A: Methods

    1992

  58. [67]

    V ariational Monte Carlo As explained in the main text, Variational Monte Carlo aims at minimizing the expectation value of the hamilto- nian under a variational state |ψθ⟩: Eθ = ⟨ψθ| H |ψθ⟩ ⟨ψθ|ψθ⟩ . By inserting the sum of projector along each element of the orthogonal basis...

  59. [68]

    The goal will be to minimize the infidelity I = 1 − F

    Infidelity optimization The fidelity between two pure quantum states is de- fined as: F (|ψ⟩ , |ϕ⟩) = ⟨ψ|ϕ⟩ ⟨ϕ|ψ⟩ ⟨ψ|ψ⟩ ⟨ϕ|ϕ⟩ , where |ψ⟩ will be identified to the variational state, and |ϕ⟩ to the target state. The goal will be to minimize the infidelity I = 1 − F. Following ...

  60. [69]

    We start by defining the SNIS estimator, then computes its variance and bias

    The self-normalized Importance Sampling estimator In this section, we review the mathematical aspects of the self-normalized Importance Sampling estimator (SNIS). We start by defining the SNIS estimator, then computes its variance and bias. Afterwards, we derive the derivative...

  61. [70]

    We will denote qSN IS opt (x) = 1 Np PNp i=1 qi opt(x) the average of the above distributions, and qα a variational family of distribution

    Alternative definitions of the objective function As mentioned in the main text, one knows analytically that the distribution minimizing the variance of each gra- dient component is: qi opt(x) ∝ |ψ(x)|2(x)|fi(x) − Fi|. We will denote qSN IS opt (x) = 1 Np PNp i=1 qi opt(x) the...

  62. [71]

    J1-J2 results a. ViT architecture The Vision Transformer, as introduced in [53], is a deep neural network architecture originally applied to Com- 0.0 0.5 1.0 1.5 2.0 α 10−3 10−2 LIS 2 4 6 8 DKL LIS DKL(qα|qIS opt) DKL(qα|qSN IS opt ) FIG. 7. Comparison of several objective for...

    2000

  63. [72]

    Neural network backflow We use an NQS with the same architecture as the one used in Ref

    Ab-initio quantum chemistry a. Neural network backflow We use an NQS with the same architecture as the one used in Ref. [6] and Ref. [29]. For results on Li 2O, we use 2 hidden layers, and 256 hidden units. For N 2, the number of hidden units is reduced to 16. To improve the r...

  64. [73]

    Infidelity minimization We employ the same hyperparameters as appendix F of Ref. [24]. As an ansatz, we use a convolutional neural network with 4 layers, a fixed kernel size of 3, and 10 channels in each layer. The diagonal shift is set auto- matically with the PI controller d...

  65. [74]

    For details about the notation and derivation, please refer to the main text and later appendices

    Equation summary (implementation aid) In this section we give all the necessary equations for the reader to implement the method presented in this paper. For details about the notation and derivation, please refer to the main text and later appendices. a. VMC We aim at minimiz...

Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.