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Hamiltonian reconstruction as metric for variational studies

by Kevin Zhang, Samuel Lederer, Kenny Choo, Titus Neupert, Giuseppe Carleo, and Eun-Ah Kim

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Submission summary

Authors (as registered SciPost users): Kevin Zhang
Submission information
Preprint Link: scipost_202202_00044v2  (pdf)
Date accepted: 2022-07-05
Date submitted: 2022-06-20 04:34
Submitted by: Zhang, Kevin
Submitted to: SciPost Physics
Ontological classification
Academic field: Physics
  • Condensed Matter Physics - Theory
  • Condensed Matter Physics - Computational
Approaches: Theoretical, Computational


Variational approaches are among the most powerful techniques to approximately solve quantum many-body problems. These encompass both variational states based on tensor or neural networks, and parameterized quantum circuits in variational quantum eigensolvers. However, self-consistent evaluation of the quality of variational wavefunctions is a notoriously hard task. Using a recently developed Hamiltonian reconstruction method, we propose a multi-faceted approach to evaluating the quality of neural-network based wavefunctions. Specifically, we consider convolutional neural network (CNN) and restricted Boltzmann machine (RBM) states trained on a square lattice spin-1/2 J1-J2 Heisenberg model. We find that the reconstructed Hamiltonians are typically less frustrated, and have easy-axis anisotropy near the high frustration point. In addition, the reconstructed Hamiltonians suppress quantum fluctuations in the large J2 limit. Our results highlight the critical importance of the wavefunction's symmetry. Moreover, the multi-faceted insight from the Hamiltonian reconstruction reveals that a variational wave function can fail to capture the true ground state through suppression of quantum fluctuations.

Published as SciPost Phys. 13, 063 (2022)

Author comments upon resubmission

Dear Editor,

We thank the editor for organizing the review of our manuscript and are grateful to the referees for their valuable comments which have led us to improve the clarity of several key points of our work.
We are happy to see that all of the reviewers find our work to be of interest to the general community, and to be built upon a solid foundation.
In general, the referees hoped for more clarification for our claims.
We have taken this opportunity to address these concerns, and we believe that this has resulted in a much improved manuscript.

Below, we have compiled the referees' reports and our responses.
The changes to the text and Supplemental Material have been marked in red in the documents.

Referee 1
To see this idea to fruition, you would want to develop more "actionable" insights from their method, e.g. by using the analysis to develop new variational ansatzes. (Could be future work.)
-> We agree with the referee, and we have added a comment that developing the ansatz to improve the dJ2 or J3 reconstruction could be future work to the beginning of page 4.

Referee 2
Clarify/strengthen the argumenation in section "verification of reconstructed hamitlonian"
-> We have rewritten this section with more clarity on what our results imply.
Specifically, we have switched to the RBM wavefunctions (which more clearly display the improvement of the reconstructed Hamiltonians over the original, although the CNN wavefunctions also havethe same improvements). We show that the variance of the reconstructed Hamiltonian is improved by the same order of magnitude as the reconstructed Hamiltonian parameters, which is to be expected from the reconstruction space we use.
However, since the variance should not be used as the sole metric for wavefunction quality, as we argue in the article, we also show the improvement in wavefunction fidelity to the ground state by using the reconstructed Hamiltonian.

Incorporate part of Supplementary Materials (especially in sec. II) in the body of the article to make it self contained.
-> We agree with the referee that the move would help to make the article self contained, and we have done so accordingly.

List of changes

The changes to the text and Supplemental Material have been marked in red in the documents.

Specifically, the changes we have made to the manuscript are as follows:
- We have moved section II.C of the Supplemental Material to the main text in page 3 to make the text self-contained. We have also rewritten this section to improve the clarity and strength of our argumentation.
- We have added a comment that improving the ansatz to improve the dJ2 or J3 reconstruction could be future work to the beginning of page 4

Submission & Refereeing History

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Resubmission scipost_202202_00044v2 on 20 June 2022

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