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Universal Performance Gap of Neural Quantum States Applied to the HofstadterBoseHubbard Model
by Eimantas Ledinauskas, Egidijus Anisimovas
This is not the latest submitted version.
Submission summary
Authors (as registered SciPost users):  Eimantas Ledinauskas 
Submission information  

Preprint Link:  https://arxiv.org/abs/2405.01981v2 (pdf) 
Date submitted:  20240724 12:42 
Submitted by:  Ledinauskas, Eimantas 
Submitted to:  SciPost Physics 
Ontological classification  

Academic field:  Physics 
Specialties: 

Approaches:  Theoretical, Computational 
Abstract
Neural Quantum States (NQS) have demonstrated significant potential in approximating ground states of manybody quantum systems, though their performance can be inconsistent across different models. This study investigates the performance of NQS in approximating the ground state of the HofstadterBoseHubbard (HBH) model, a boson system on a twodimensional square lattice with a perpendicular magnetic field. Our results indicate that increasing magnetic flux leads to a substantial increase in energy error, up to three orders of magnitude. Importantly, this decline in NQS performance is consistent across different optimization methods, neural network architectures, and physical model parameters, suggesting a significant challenge intrinsic to the model. Despite investigating potential causes such as wave function phase structure, quantum entanglement, fractional quantum Hall effect, and the variational loss landscape, the precise reasons for this degradation remain elusive. The HBH model thus proves to be an effective testing ground for exploring the capabilities and limitations of NQS. Our study highlights the need for advanced theoretical frameworks to better understand the expressive power of NQS which would allow a systematic development of methods that could potentially overcome these challenges.
Author indications on fulfilling journal expectations
 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 multipronged followup work
 Detail a groundbreaking theoretical/experimental/computational discovery
 Present a breakthrough on a previouslyidentified and longstanding research stumbling block
Author comments upon resubmission
List of changes
 revised introduction section
 revised conclusion section
 changed the normalization of energy in Figure 1 as suggested by referee 1
Current status:
Reports on this Submission
Strengths
The manuscript provides an extensive numerical study on different NQS architectures, hyperparameters, optimization strategies and potential challenges for applying NQS to the HBH model, and could hence be a useful reference for the community.
Weaknesses
Despite the extensive study of potential challenges, the authors did not succeed in lowering the error in some regimes of the phase diagram. Given that many challenges identified by the authors are still a matter of current research, I think that the manuscript is still a valuable reference for the field.
Report
In their manuscript, the authors provide a detailed study of NQS approaches for the HofstadterBoseHubbard model. The manuscript is clearly written, provides useful references to previous works in the field as well as detailed information on the calculations that were done. The approaches to identify (and hopefully at some point resolve) the challenges of the NQS approach to represent this system could be a valuable reference for future works in the field. Hence, I recommend this work for publication after revising / answering the following comments and questions.
Remarks:
 Fig. 1: The authors mention that their DMRG results were obtained with a bond dimension of 8 (corresponding to a similar number of parameters as used for the NQS), which seems very small. It would be very important to see how the results compare to a MPS calculation with more parameters, requiring e.g. similar computation time as the NQS results.
 Maybe Eq. (11) could be moved into Sec. 3.3.3?
Questions:
 As far as I understand, both SR and SITE perform imaginary time evolution, but the difference between them is that different distance measures between are applied (Fubini study distance vs. overlap). Is this correct? In any case, it would be helpful for the reader if the differences / similarities would be highlighted in the manuscript.
 Concerning Sec. 4.2.1.: The authors point out that there is a difference between the statistics of the wave function elements for different alpha. Is it somewhere mentioned if the respective NQS wave functions were trained with the complete set of samples? If not, would Fig. 2 look better if it was trained with the complete set?
 Concerning the analysis of the minima: "neither the ruggedness nor the curvature of the loss landscape is the source of the performance issues observed". I have 2 questions on this comment: 1) Given the relatively high errors in Fig.1, could it be that the analyzed minimum is not the global minimum? 2) How does the minimum develop when more / less parameters are used?
Recommendation
Ask for minor revision
Report
The authors investigate the performance of Neural Quantum States (NQS) to approximate the ground state of the HofstadterBoseHubbard model, a noninteracting hardcore boson model on a 2D square lattice coupled to a perpendicular magnetic field. Using numerical experiments for small system sizes with various network and loss function setups, the authors show a consistent drop in NQS performance when approaching the strong magnetic flux regime. Despite exploring different explanations for the reason behind this, the cause for the performance degradation remains elusive.
The paper is well written, and results are clearly presented, but I believe that it is missing a guiding thread and a clear take home message for the reader. In the current form, the observation of the performance degradation is followed by a list of explanations that do not explain it. Outlining the conducted experiments in the main text is valid and important, but since these do not explain the observed phenomena, I think the details should be reported in an appendix/supplementary. Without an explanation for the performance degradation, or a proposal how to mitigate it (even if just by making the network bigger; see below), this seems not to be sufficiently engaging.
The authors state that they restrict the network size to be smaller than the Hilbert space dimension. Previous works on NQS performance [see in particular Phys. Rev. Lett. 131, 036502] have also investigated the number of required network parameters to accurately represent a given ground state. Missing a reason for the performance degradation in this case, I think it would be important to know how to mitigate it. Do I need as many NQS parameters as my hilbert space dimension to represent the ground state accurately? How does the required number of NQS parameters behave as a function of magentic flux? This would at least unveil if the NQS is unable to learn an efficient compression of the wave function at strong magnetic flux or if the observed performance drop has more to do with the optimization scheme(s).
In conclusion, the paper contains valid and well presented calculations. An important missing piece is a reason for the performance drop, or at least a discussion how to mitigate the effect by modifying e.g. network size. Since reporting on drawbacks and shortcomings of emerging methods such as NQS is important (and dearly missing in the literature), I still think the manuscript should be published with appropriate modifications.
Recommendation
Ask for major revision
Author: Eimantas Ledinauskas on 20241014 [id 4865]
(in reply to Report 1 on 20240904)
Thank you for your comments and suggestions.
We agree that it is somewhat disappointing that we did not identify the exact cause or solution for the performance decline. However, publishing negative results is also important for scientific progress, and we hope these findings will contribute to the search for a solution. The central focus of our paper is to introduce the HBH model as a new, challenging testing ground for NQS methods, which is difficult to solve and relevant to broader physics research.
Regarding your remarks about the number of parameters required to accurately represent the ground state, we find this to be an interesting question. We have added a new Figure 2 and a paragraph discussing this topic in Section 4.1.
In your report, you wrote “HofstadterBoseHubbard model, a noninteracting hardcore boson model”. Let us stress that the HBH model describes a system of interacting bosons in the limit where interactions are particularly strong. Although the interactions are not directly visible in the Hamiltonian, they are implemented by means of the hardcore constraint. In particular, a recent work [SciPost Phys 12, 095 (2022)] has demonstrated that the model is suitable to describe the fractional quantum Hall regime in lattice systems.
Author: Eimantas Ledinauskas on 20241014 [id 4866]
(in reply to Report 2 on 20241008)Thank you for your comments and suggestions. Here are our responses to the two remarks, R1 and R2 and the three questions Q1, Q2, and Q3.
[R1] We added data for DMRG with a virtual bond dimension of 24, where the MPS has approximately four times more parameters than the Hilbert space of the model being studied.
[R2] We moved Eq. 11 to Sec. 3.3.3.
[Q1] We added brief comments on the differences between SR and SITE in Sec. 3.3.2 and included a reference for a more detailed discussion.
[Q2] In all cases we trained the neural networks by using the complete set of the basis vectors in the Hilbert space.
[Q3.1] Regarding the loss landscape, there are no guarantees that the found minimum is the global one because of the local optimization scheme. By stating that “neither the ruggedness nor the curvature of the loss landscape is the source of the performance issues observed”, we mean that we do not observe any significant differences between weak and strong magnetic flux regimes when studying the loss landscape.
[Q3.2] Regarding the development of the identified minimum with different parameter counts, we note that changing the parameter count, such as by increasing the neural network width, alters the network architecture and thus changes the loss landscape in nontrivial ways. Therefore, there is no straightforward way to analyze the evolution of a specific minimum. However, Figures 1 and 2 show that the found minimum becomes deeper as the number of parameters increases.