Projective quantum Monte Carlo simulations guided by unrestricted neural network states

TL;DR

This paper demonstrates that using unrestricted neural network states as guiding functions in projective quantum Monte Carlo significantly improves efficiency, enabling polynomial scaling of computational cost even at critical points in quantum spin models.

Contribution

The study introduces the use of unrestricted Boltzmann machine states as guiding functions in PQMC, showing enhanced accuracy and efficiency over previous methods with restricted models.

Findings

Unrestricted neural network states accurately approximate ground states with few parameters.

Importance sampling with these states reduces systematic bias in PQMC.

Computational cost scales polynomially with system size at criticality.

Abstract

We investigate the use of variational wave-functions that mimic stochastic recurrent neural networks, specifically, unrestricted Boltzmann machines, as guiding functions in projective quantum Monte Carlo (PQMC) simulations of quantum spin models. As a preliminary step, we investigate the accuracy of such unrestricted neural network states as variational Ans\"atze for the ground state of the ferromagnetic quantum Ising chain. We find that by optimizing just three variational parameters, independently on the system size, accurate ground-state energies are obtained, comparable to those previously obtained using restricted Boltzmann machines with few variational parameters per spin. Chiefly, we show that if one uses optimized unrestricted neural network states as guiding functions for importance sampling the efficiency of the PQMC algorithms is greatly enhanced, drastically reducing the most relevant systematic bias, namely that due to the finite random-walker population. The scaling of the computational cost with the system size changes from the exponential scaling characteristic of PQMC simulations performed without importance sampling, to a polynomial scaling, even at the ferromagnetic quantum critical point. The important role of the protocol chosen to sample hidden-spins configurations, in particular at the critical point, is analyzed. We discuss the implications of these findings for what concerns the problem of simulating adiabatic quantum optimization using stochastic algorithms on classical computers.