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Continuous-variable neural-network quantum states and the quantum rotor model

by James Stokes, Saibal De, Shravan Veerapaneni, Giuseppe Carleo

Submission summary

Authors (as registered SciPost users): James Stokes
Submission information
Preprint Link: https://arxiv.org/abs/2107.07105v1  (pdf)
Code repository: https://github.com/shravanvn/cnqs
Date submitted: 2022-03-09 20:40
Submitted by: Stokes, James
Submitted to: SciPost Physics Core
Ontological classification
Academic field: Physics
Specialties:
  • Condensed Matter Physics - Computational
  • Quantum Physics
Approaches: Theoretical, Computational

Abstract

We initiate the study of neural-network quantum state algorithms for analyzing continuous-variable lattice quantum systems in first quantization. A simple family of continuous-variable trial wavefunctons is introduced which naturally generalizes the restricted Boltzmann machine (RBM) wavefunction introduced for analyzing quantum spin systems. By virtue of its simplicity, the same variational Monte Carlo training algorithms that have been developed for ground state determination and time evolution of spin systems have natural analogues in the continuum. We offer a proof of principle demonstration in the context of ground state determination of a stoquastic quantum rotor Hamiltonian. Results are compared against those obtained from partial differential equation (PDE) based scalable eigensolvers. This study serves as a benchmark against which future investigation of continuous-variable neural quantum states can be compared, and points to the need to consider deep network architectures and more sophisticated training algorithms.

Current status:
Awaiting resubmission

Reports on this Submission

Anonymous Report 2 on 2022-5-11 (Invited Report)

Strengths

1. Attempts to make a step in a natural direction and generalize neural-network quantum states on the case of continuous models.

Weaknesses

1. Only very small systems considered (up to 7 particles).
2. The analysis of truly continuous systems is limited by the 4-particle case.
3. The overly mathematical style of writing seems unneeded to convey the main messages.

Report

This paper addresses a natural question of how neural-network quantum states can be used to simulate continuous quantum systems, and if employing them is beneficial in this context.

Before the paper can be reconsidered for publication in SciPost, it has to be substantially improved. In general, I tend to agree with both points raised by the first referee.

First of all, the adopted style of writing that heavily exploits mathematical notations and terminology is indeed very confusing. If the authors intend to reach out to the community of mathematical physicists, a more specialized journal than SciPost would serve this purpose better. If, however, the main audience is their peers from the computational quantum physics domain, I would strongly advise to simplify notations and rewrite the manuscript in the more typical "physical" way, or, instead, consider a physical situation that naturally requires using the Riemannian geometry approach (a chain of quantum rotors is not the best example in this sense).

Secondly, from the study of seven particles, it is hard to judge whether the method scales nicely to larger systems, and what accuracy it gives in comparison with other approaches.
It is not necessary to go for really huge number of rotors, but a system of ~30-40 particles should be considered.

Besides that, I would be happy to see a more detailed analysis of truly continuous systems. In the language of the paper, it would mean that fully connected graphs need to be considered, so that every two particles on the manifold interact with each other. If my understanding is correct, now it is done only for four particles, but how do the error and elapsed time scale with the number of rotors in the fully connected case? Is it possible to go up to 10 sites to check this?

Requested changes

1. Consider larger systems of at least 30-40 rotors.
2. Study the accuracy scaling for fully connected graphs.
3. Change the writing style.

  • validity: good
  • significance: low
  • originality: good
  • clarity: ok
  • formatting: -
  • grammar: -

Anonymous Report 1 on 2022-5-8 (Invited Report)

Strengths

None at the moment

Weaknesses

Calculations do not justify the motivations in the introduction.

Calculations are limited to very few sites.

Report

In this work, Stokes and co-workers would like to address quantum problems with continuous degrees of freedom by using neural-network variational states. I am a bit confused about the presentation and the results of the paper.

First of all, the introduction discuss perspectives about Hamiltonians with ``non-Euclidean structure, as the hyperbolic lattices'' or about describing states of the Hilbert space which are invariant or equivalent to group invariances of the Hamiltonian. All the discussion about Eq.(1) is also in this line. However, at the end of the day, they finally focus on the quantum rotor model in one dimension. Moreover, if I understand correctly, they considered a very few sites (mainly 4, at most 7).

In addition, the quantum rotor model can be easily described within a discrete Hilbert space, since it is equivalent to the Bose-Hubbard model. I would be curious to see an example in which there is a purely continuous Hilbert space. By the way, they use mathematical symbols that are unnecessarily complicated for what they do (a one-dimensional lattice). In this regard, the first two pages are completely off with respect to what is shown in section 3.

For these reasons, I do not think that the paper is suitable for a publication. Even in the idea of considering the quantum rotor model (which I do not fully like), it would be fair to, first of all, modify the introduction, to have something suitable for the actual calculations. In additon, I would ask for having calculations on larger system sizes, in order to clearly show that the method is competitive with other ones.

Requested changes

Re-write the initial part of the paper.

Do calculations on larger sizes.

  • validity: poor
  • significance: poor
  • originality: good
  • clarity: poor
  • formatting: -
  • grammar: -

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