SciPost Submission Page
Instability of explicit time integration for strongly quenched dynamics with neural quantum states
by Hrvoje Vrcan, Johan H. Mentink
Submission summary
| Authors (as registered SciPost users): | Hrvoje Vrcan |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2507.17421v2 (pdf) |
| Code repository: | https://github.com/HVrcan/lattice_nqs |
| Date submitted: | Dec. 24, 2025, 12:14 a.m. |
| Submitted by: | Hrvoje Vrcan |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| Academic field: | Physics |
| Specialties: |
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| Approach: | Computational |
Abstract
Neural quantum states have recently demonstrated significant potential for simulating quantum dynamics beyond the capabilities of existing variational ansätze. However, studying strongly driven quantum dynamics with neural networks has proven challenging so far. Here, we focus on assessing several sources of numerical instabilities that can appear in the simulation of quantum dynamics based on the time-dependent variational principle (TDVP) with the computationally efficient explicit time integration scheme. Focusing on the restricted Boltzmann machine architecture, we compare solutions obtained by TDVP with analytical solutions and implicit methods as a function of the quench strength. Interestingly, we uncover a quenching strength that leads to a numerical breakdown in the absence of Monte Carlo noise, despite the fact that physical observables don't exhibit irregularities. This breakdown phenomenon appears consistently across several different TDVP formulations, even those that eliminate small eigenvalues of the Fisher matrix or use geometric properties to recast the equation of motion. We provide evidence that the nature of the instability stems from stiffness of the dynamics of the variational parameters, despite the absence of stiffness in the exact quantum dynamics. We conclude that alternative methods need to be developed to leverage the computational efficiency of explicit time integration of the TDVP equations for simulating strongly nonequilibrium quantum dynamics with neural-network quantum states.
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- Provide a novel and synergetic link between different research areas.
- Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work
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- Present a breakthrough on a previously-identified and long-standing research stumbling block
Author comments upon resubmission
We hereby resubmit the manuscript for our paper, titled “Instability of explicit time integration for strongly quenched dynamics with neural quantum states.” The referee finds our results interesting and reasonable, but requests more numerical evidence, which we provide in the resubmitted manuscript. In particular, addressing the referee comments helped us identify that the nature of the instability can be mathematically understood as a stiff problem for the variational parameters, despite the fact that the exact quantum dynamics is not stiff. Evidence for this interpretation is provided by an analysis using an adaptive integrator. This finding is now included in the main text, including a new figure. We added several paragraphs in the Discussion section commenting on other possible origins of the numerical instability unmentioned in the original submission, as well as several appendices with figures supporting the analysis presented. Importantly, our main conclusion that alternative methods need to be developed to leverage the computational efficiency of explicit time integration of the TDVP equations remains still valid. We thank the referee for the constructive comments, which added valuable insights to our paper. We are confident that these changes will make the manuscript adequate for publication.
Please find our detailed answer to the referee comments and the list of changes on the submission page.
Sincerely,
Hrvoje Vrcan
List of changes
- Added a sentence to the Abstract: “We provide evidence that the nature of the instability stems from stiffness of the dynamics of the variational parameters, despite the absence of stiffness in the exact quantum dynamics.”
- Added one sentence to paragraph 2 of the Introduction: “For example, in contrast to solving a continuous-time ODE of variational parameters, methods introduced in [10,13,14,22,24] solve an optimization problem at each step.” Rearranged some of the following sentences.
- Changes to the Discussion section: a. Paragraph 3: i. added a sentence with a reference to Appendix D, ii. removed a sentence about adaptive integration, iii. removed a sentence about stiffness. b. Added a new paragraph 4 about the stiffness of the equations of motion and the RK45 adaptive integrator. Other paragraphs have been shifted forward. c. Added a new Figure 5, with the results of the RK45 adaptive integrator. d. Added a sentence at the end of paragraph 5 about other neural network models. e. Added a new paragraph 6 about different ways to calculate quantum expectation values, and a previously reported biasing problem. f. Added a new paragraph 8, commenting on Differential Algebraic Equations (DAE-s).
- Changed several sentences in the Conclusion to address stiffness and the adaptive integrator.
- Added one sentence at the end of the Acknowledgements.
- Added Appendix D: Comparison of variational parameters a. Includes new Figure 6.
- Added Appendix E: RK45 adaptive integrator.
- Added Appendix F: Other neural network architectures. a. Includes new Figure 7.
- Added Appendix G: psi-logpsi formulations a. Includes new Figure 8.