# Self-learning projective quantum Monte Carlo simulations guided by restricted Boltzmann machines

## TL;DR

This paper introduces a self-learning projective quantum Monte Carlo method guided by restricted Boltzmann machines, which adaptively optimizes the wave function during simulation, eliminating the need for separate variational steps.

## Contribution

The novel approach integrates machine learning into PQMC, enabling adaptive wave function optimization directly during simulations, improving accuracy and efficiency.

## Key findings

- Accurate ground state simulations of the quantum Ising chain
- High agreement with Jordan-Wigner and loop quantum Monte Carlo results
- Eliminates separate variational optimization step

## Abstract

The projective quantum Monte Carlo (PQMC) algorithms are among the most powerful computational techniques to simulate the ground state properties of quantum many-body systems. However, they are efficient only if a sufficiently accurate trial wave function is used to guide the simulation. In the standard approach, this guiding wave function is obtained in a separate simulation that performs a variational minimization. Here we show how to perform PQMC simulations guided by an adaptive wave function based on a restricted Boltzmann machine. This adaptive wave function is optimized along the PQMC simulation via unsupervised machine learning, avoiding the need of a separate variational optimization. As a byproduct, this technique provides an accurate ansatz for the ground state wave function, which is obtained by minimizing the Kullback-Leibler divergence with respect to the PQMC samples, rather than by minimizing the energy expectation value as in standard variational optimizations. The high accuracy of this self-learning PQMC technique is demonstrated for a paradigmatic sign-problem-free model, namely, the ferromagnetic quantum Ising chain, showing very precise agreement with the predictions of the Jordan-Wigner theory and of loop quantum Monte Carlo simulations performed in the low-temperature limit.