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Improving neural network performance for solving quantum sign structure

Xiaowei Ou, Tianshu Huang, Vidvuds Ozolins

TL;DR

This work tackles learning ground states of non-stoquastic Hamiltonians with neural quantum states by decoupling the optimization of amplitude and phase. It introduces time-step adjusted stochastic reconfiguration, leveraging a larger imaginary-time step for phase updates to balance gradient contributions and stabilize learning. The method, including MinSR and a preconditioned variant using a modified operator, achieves state-of-the-art accuracy on the $J_1$-$J_2$ Heisenberg model on a $6\times 6$ lattice, even near maximal frustration where sign structure is challenging. By combining symmetry adaptation, robust optimization, and phase–amplitude decoupling, the approach improves convergence and scalability to more complex non-stoquastic problems, with implications for variational Monte Carlo and broader neural-network quantum-state applications.

Abstract

Neural quantum states have emerged as a widely used approach to the numerical study of the ground states of non-stoquastic Hamiltonians. However, existing approaches often rely on a priori knowledge of the sign structure or require a separately pre-trained phase network. We introduce a modified stochastic reconfiguration method that effectively uses differing imaginary time steps to evolve the amplitude and phase. Using a larger time step for phase optimization, this method enables a simultaneous and efficient training of phase and amplitude neural networks. The efficacy of our method is demonstrated on the Heisenberg J_1-J_2 model.

Improving neural network performance for solving quantum sign structure

TL;DR

This work tackles learning ground states of non-stoquastic Hamiltonians with neural quantum states by decoupling the optimization of amplitude and phase. It introduces time-step adjusted stochastic reconfiguration, leveraging a larger imaginary-time step for phase updates to balance gradient contributions and stabilize learning. The method, including MinSR and a preconditioned variant using a modified operator, achieves state-of-the-art accuracy on the - Heisenberg model on a lattice, even near maximal frustration where sign structure is challenging. By combining symmetry adaptation, robust optimization, and phase–amplitude decoupling, the approach improves convergence and scalability to more complex non-stoquastic problems, with implications for variational Monte Carlo and broader neural-network quantum-state applications.

Abstract

Neural quantum states have emerged as a widely used approach to the numerical study of the ground states of non-stoquastic Hamiltonians. However, existing approaches often rely on a priori knowledge of the sign structure or require a separately pre-trained phase network. We introduce a modified stochastic reconfiguration method that effectively uses differing imaginary time steps to evolve the amplitude and phase. Using a larger time step for phase optimization, this method enables a simultaneous and efficient training of phase and amplitude neural networks. The efficacy of our method is demonstrated on the Heisenberg J_1-J_2 model.

Paper Structure

This paper contains 6 sections, 26 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Neural quantum states
  3. Stochastic Reconfiguration
  4. Time-step adjusted optimization
  5. Discussion
  6. Symmetrized wave function

Figures (3)

  • Figure 1: Neural network architecture.
  • Figure 2: Comparison of energy learning curves for (a) $J_2/J_1=0$ and (b) $J_2/J_1=0.5$. $\tilde{\epsilon}$ method (SR) refers to Eq. (\ref{['eq:delta_theta_modified_SR']}) and $\tilde{\epsilon}$ method (MinSR) refers to Eq. (\ref{['eq:delta_theta_modified_MinSR']}) with $m=4$, while $\tilde{O}$ method (SR) represents Eq. (\ref{['eq:delta_theta_modified_SR_twotilde']}) and $\tilde{O}$ method (MinSR) represents Eq. (\ref{['eq:delta_theta_modified_MinSR_twotilde']}) with $m=11$.
  • Figure 3: Cumulative distribution function of phase relative to MSR for $J_2/J_1=0$ and $J_2/J_1=0.5$.