  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 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.
