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Universal Performance Gap of Neural Quantum States Applied to the Hofstadter-Bose-Hubbard Model
by Eimantas Ledinauskas, Egidijus Anisimovas
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Submission summary
| Authors (as registered SciPost users): | Eimantas Ledinauskas |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2405.01981v1 (pdf) |
| Date submitted: | 2024-05-06 06:40 |
| Submitted by: | Ledinauskas, Eimantas |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approaches: | Theoretical, Computational |
Abstract
Neural Quantum States (NQS) have demonstrated significant potential in approximating ground states of many-body 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 Hofstadter-Bose-Hubbard (HBH) model, a boson system on a two-dimensional 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 fundamental 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 multi-pronged follow-up work
- Detail a groundbreaking theoretical/experimental/computational discovery
- Present a breakthrough on a previously-identified and long-standing research stumbling block
Current status:
Reports on this Submission
Report
The manuscript is concerned with questions related to the training of Neural Quantum States (NQS), a recent numerical approach for approximating ground states of quantum many-body Hamiltonians. The authors focus on the Hofstadter-Bose-Hubbard (HBH) model, a non-interacting boson system on a two-dimensional square lattice with a perpendicular magnetic field. Using multi-layer perceptron and convolutional neural network architectures, the authors perform numerical experiments in three different setups (Variational Monte Carlo optimization of the energy, supervised imaginary time evolution and supervised learning of exact solution) to consistently show a drop in NQS performance in the strong magnetic flux regime. To uncover this drop in performance, the authors analyzed possible explanations related to the statistics of the wave function elements, the complex phase structure, quantum entanglement, the fractional quantum Hall effect, and the variational loss landscape. However, the precise reason for this degradation remain elusive.
Overall, the paper is well written and the results are clearly presented. However, I don't think that the paper fulfill the journal expectation indicated from the authors, namely, to `Open a new pathway in an existing or a new research direction, with clear potential for multi-pronged follow-up work'. Specifically, I question the choice of the HBH model as a particularly effective testing ground for NQS. With the architectures and optimization protocols outlined in the paper, similar conclusions could likely be reached for many other physical models. For instance, similar results to those presented in Fig. 1 (left panel) of the manuscript could be achieved by optimizing a multi-layer perceptron within the Variational Monte Carlo framework on the J1-J2 Heisenberg model on a 4x4 square lattice. Using Stochastic Reconfiguration and a relatively small batch size (e.g., 512, as considered in the present work), the final relative error would worsen by orders of magnitude when increasing the frustration ratio to J2/J1. However, it is incorrect to draw conclusions such as "there are fundamental limitations of NQS in describing the physics of the 2D J1−J2 Heisenberg model." On the contrary, NQS-based architectures have achieved state-of-the-art ground state energies for this system, surpassing other traditional methods (see for example arXiv:2302.01941, Phys. Rev. X 11, 031034, arXiv:2310.05715). Furthermore, also for bosonic models, NQS can achieve state-of-the-art energies on the 2D Bose-Hubbard model (see https://arxiv.org/pdf/2404.07869). The latter is an interacting model and consequently more complicated than the one described in the paper under review.
Summarizing, while the manuscript contains valid and well-presented calculations, the conclusion that NQS are unsuitable for studying the HBH model is not appropriate. The authors should revise the discussion of their results in the context of their particular setup, avoiding generalizations about the limitations NQS.
As a minor comment, in Fig. 1 (left panel), the y-axis label currently shows E_NQS - E_ED. It would be more informative to report the relative error, abs((E_NQS - E_ED)/E_ED), instead.
Recommendation
Ask for major revision
Author: Eimantas Ledinauskas on 2024-07-08 [id 4607]
(in reply to Report 1 on 2024-06-28)Thank you for your comments. We acknowledge that our text in the introduction and conclusion sections could be improved to more clearly reflect our intended claims, and we will gladly revise it. Below are our replies to some specific statements.
The referee writes:
Our response: 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 hard-core 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.
The referee writes:
Our response:
It is true that a drop in NQS performance is also observed in the J1-J2 model, which we mentioned in our introduction. However, it is important to also consider complementary models and gather more such challenging examples to better understand the characteristics of the ground state that lead to this decline. Moreover, our results suggest that the reasons for performance decline in the HBH model might differ from those in the J1-J2 model. For example, unlike [SciPost Phys 10, 147 (2021)], which analyzed the J1-J2 model, we do not find evidence that the decline in the HBH model is due to a rugged variational loss landscape.
The referee writes:
Our response: We do not make such a general conclusion (or at least do not intend to). In our conclusions, we specifically write: "It remains uncertain whether this challenge can be addressed through some modifications or if it represents a more fundamental limitation of NQS, potentially requiring problem-specific adjustments.". In this work, we demonstrate a performance decline of several orders of magnitude, which cannot be solved by the various adjustments we described. We think it is important to identify and analyze such cases, even if NQS can achieve better performance than other state-of-the-art methods, because this might lead to realizations on how to improve NQS architectures or optimization procedures. These challenging models are analogous to challenging datasets in the machine learning field, like ImageNet, which historically catalyzed the development of breakthroughs.
We hope that our replies and updated text will change the referee's opinion about the validity and significance of our work.