Role of stochastic noise and generalization error in the time propagation of neural-network quantum states

TL;DR

This paper investigates how stochastic noise affects the stability and accuracy of neural-network quantum states during time evolution, proposing regularization and validation tools to improve simulation fidelity in out-of-equilibrium quantum systems.

Contribution

It introduces a regularization approach and a validation-set diagnostic for enhancing the stability of neural-network quantum state simulations during time propagation.

Findings

Regularization significantly improves stability in neural-network quantum state evolution.

Validation-set diagnostics help optimize regularization hyperparameters.

Stable and accurate dynamics are achievable with proper regularization.

Abstract

Neural-network quantum states (NQS) have been shown to be a suitable variational ansatz to simulate out-of-equilibrium dynamics in two-dimensional systems using time-dependent variational Monte Carlo (t-VMC). In particular, stable and accurate time propagation over long time scales has been observed in the square-lattice Heisenberg model using the Restricted Boltzmann machine architecture. However, achieving similar performance in other systems has proven to be more challenging. In this article, we focus on the two-leg Heisenberg ladder driven out of equilibrium by a pulsed excitation as a benchmark system. We demonstrate that unmitigated noise is strongly amplified by the nonlinear equations of motion for the network parameters, which causes numerical instabilities in the time evolution. As a consequence, the achievable accuracy of the simulated dynamics is a result of the interplay between network expressiveness and measures required to remedy these instabilities. We show that stability can be greatly improved by appropriate choice of regularization. This is particularly useful as tuning of the regularization typically imposes no additional computational cost. Inspired by machine learning practice, we propose a validation-set based diagnostic tool to help determining optimal regularization hyperparameters for t-VMC based propagation schemes. For our benchmark, we show that stable and accurate time propagation can be achieved in regimes of sufficiently regularized variational dynamics.