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Role of stochastic noise and generalization error in the time propagation of neural-network quantum states

by D. Hofmann, G. Fabiani, J. H. Mentink, G. Carleo, M. A. Sentef

This Submission thread is now published as

Submission summary

Authors (as registered SciPost users): Giuseppe Carleo · Damian Hofmann · Michael Sentef
Submission information
Preprint Link: scipost_202111_00037v1  (pdf)
Date accepted: 2022-04-29
Date submitted: 2021-11-19 16:58
Submitted by: Hofmann, Damian
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Neural-network quantum states (NQS) have been shown to be a suitable variational ansatz to simulate out-of-equilibrium dynamics in two-dimensional systems using time-dependent variational Monte Carlo (t-VMC). In particular, stable and accurate time propagation over long time scales has been observed in the square-lattice Heisenberg model using the Restricted Boltzmann machine architecture. However, achieving similar performance in other systems has proven to be more challenging. In this article, we focus on the two-leg Heisenberg ladder driven out of equilibrium by a pulsed excitation as a benchmark system. We demonstrate that unmitigated noise is strongly amplified by the nonlinear equations of motion for the network parameters, which causes numerical instabilities in the time evolution. As a consequence, the achievable accuracy of the simulated dynamics is a result of the interplay between network expressiveness and measures required to remedy these instabilities. We show that stability can be greatly improved by appropriate choice of regularization. This is particularly useful as tuning of the regularization typically imposes no additional computational cost. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining optimal regularization hyperparameters for t-VMC based propagation schemes. For our benchmark, we show that stable and accurate time propagation can be achieved in regimes of sufficiently regularized variational dynamics.

Published as SciPost Phys. 12, 165 (2022)

Author comments upon resubmission

Dear Editor,

We wish to thank you for handling the review of our submission and for
giving us the opportunity of resubmitting our revised manuscript.

Furthermore, we thank both Referees for their careful reading of our
article and their specific and thoughtful feedback and suggestions,
which we have taken into account in this resubmission.

We have provided a detailed response to the individual points of each
report and the changes we have done as a result in an author comment
on the respective report.

We believe that with these improvements, our manuscript is now ready for
publication in SciPost Physics.

Yours sincerely,
on behalf of all authors,

Damian Hofmann

List of changes

In the revised manuscript uploaded here, we have highlighted textual changes (except for minor corrections or changes in notation) in blue color and provide a brief summary below. Please see our replies to the two referee reports for more details on specific changes.

- Parts of the abstract and introduction have been updated based on referee feedback.
- The initial presentation of the model (around Eqs. 1 and 2) has been improved.
- More detailed formulas for t-VMC propagation have been moved from Appendix B to Sect. 2 (Eqs. 5-7).
- The introductory sentences of Sect. 3 have been clarified.
- Details on time step and MC sampling have been added to Sect. 3.1.
- A new Figure 5 has been added (see our reply to report 1, requested change 3)
and is now described in Sect. 3.1.
- The symbol for the eigenvalues of the quantum Fisher matrix in Sect. 3.2 has been changed.
- Results and corresponding discussion of the validation error as a function of Monte Carlo sample size have been moved to a new Appendix E in order to provide better focus on our primary message in the main text.
- We have updated Appendix C to clarify our supervised learning scheme (see our reply to report 2, requested change 3).
- Several arXiv references have been updated to the published version.

Reports on this Submission

Anonymous Report 1 on 2022-4-11 (Invited Report)

  • Cite as: Anonymous, Report on arXiv:scipost_202111_00037v1, delivered 2022-04-11, doi: 10.21468/SciPost.Report.4904


See my previous report (Report 2)




The authors have addressed all issues raised in my previous report in a satisfactory manner. The added data and improved presentation render this work a comprehensive and valuable study illuminating important technical aspects of t-VMC with neural-network quantum states. The new cross-validation perspective on the solution of the time-dependent variational principle opens new routes for future adaptive regularization schemes that will be crucial for numerically stable simulations of time evolution with neural-network quantum states.

The manuscript furthermore fulfils the general acceptance criteria of SciPost Physics, and, therefore, I recommend it for publication in its current form.

Requested changes


  • validity: high
  • significance: high
  • originality: top
  • clarity: high
  • formatting: excellent
  • grammar: perfect

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