Geometry of learning neural quantum states

TL;DR

This paper analyzes the learning dynamics of neural network quantum states, revealing how spectral properties and entanglement relate to the stability and phase transitions in quantum many-body problems.

Contribution

It uncovers hidden details of the learning landscape, including spectral behavior and entanglement measures, advancing understanding of neural quantum state optimization.

Findings

Quantum Fisher matrix spectrum exhibits universal initial dynamics.

Converged spectrum changes across phase transitions.

Low-entanglement modes have larger eigenvalues, indicating stable ground state representations.

Abstract

Combining insights from machine learning and quantum Monte Carlo, the stochastic reconfiguration method with neural network Ansatz states is a promising new direction for high-precision ground state estimation of quantum many-body problems. Even though this method works well in practice, little is known about the learning dynamics. In this paper, we bring to light several hidden details of the algorithm by analyzing the learning landscape. In particular, the spectrum of the quantum Fisher matrix of complex restricted Boltzmann machine states exhibits a universal initial dynamics, but the converged spectrum can dramatically change across a phase transition. In contrast to the spectral properties of the quantum Fisher matrix, the actual weights of the network at convergence do not reveal much information about the system or the dynamics. Furthermore, we identify a new measure of correlation in the state by analyzing entanglement in eigenvectors. We show that, generically, the learning landscape modes with least entanglement have largest eigenvalue, suggesting that correlations are encoded in large flat valleys of the learning landscape, favoring stable representations of the ground state.