Symmetry Constraints Regularize Neural Quantum State Learning

Neural quantum states (NQS) are powerful tools for representing many-body wavefunctions, but they often suffer from optimization challenges such as redundant parameters and poorly conditioned landscapes. This paper investigates whether embedding physical Hamiltonian symmetries directly into the variational parameterization can regularize the learning problem and improve optimization efficiency.

The authors propose a framework for Boltzmann-family NQS that enforces symmetries by tying local Pauli-Z generators along physical geometric orbits (e.g., translations, reflections, and spin-flips). By analytically collapsing the trainable coefficient space prior to optimization, the authors reduce the number of parameters and eliminate redundant directions. To quantify the impact of this approach, they introduce a geometric metric based on the Jacobian and Hessian of the optimization landscape, which evaluates the fraction of the physically accessible state space corresponding to low-energy solutions.

Testing their approach on transverse-field Ising (TFIM) and XXZ spin chains, the authors demonstrate that symmetry compilation excises the vast majority of parameters—compressing thousands of parameters down to tens in large TFIM systems—while maintaining ground-state accuracy. The geometric diagnostics reveal that symmetry imposition prunes auxiliary flat directions and concentrates the reachable state space around low-energy solutions. This results in substantial runtime accelerations and a more favorable, target-aware optimization geometry.

This work provides a systematic way to improve the trainability and scalability of NQS models. By hard-wiring physical invariants into the model structure, researchers can reduce the computational cost of training while simultaneously improving the stability and convergence of the optimization process. This approach offers a path toward more efficient simulation of complex many-body quantum systems.