Universal Performance Gap of Neural Quantum States Applied to the Hofstadter-Bose-Hubbard Model

TL;DR

This paper examines the limitations of Neural Quantum States in modeling the Hofstadter-Bose-Hubbard system, revealing a universal performance gap that worsens with increased magnetic flux, highlighting fundamental challenges in current NQS approaches.

Contribution

It uncovers a universal performance degradation of NQS in the HBH model across various methods and parameters, emphasizing the need for new theoretical insights.

Findings

Performance declines exponentially with magnetic flux

Degradation is consistent across different neural network architectures

The exact cause of performance gap remains unknown

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, an interacting 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 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.