Green Function Monte Carlo with Stochastic Reconfiguration: an effective remedy for the sign problem disease

TL;DR

This paper introduces a Monte Carlo method with stochastic reconfiguration to systematically correct the fixed node approximation, effectively mitigating the sign problem in quantum simulations and enabling stable, accurate ground state calculations.

Contribution

The authors develop a systematic correction scheme for the fixed node approximation using stochastic reconfiguration, improving the sign problem handling in quantum Monte Carlo methods.

Findings

The method stabilizes the Monte Carlo average sign for large imaginary times.

It accurately reproduces exact diagonalization results on finite lattices.

Parameter dependence in the stochastic technique is easily controllable.

Abstract

A recent technique, proposed to alleviate the ``sign problem disease'', is discussed in details. As well known the ground state of a given Hamiltonian $H$ can be obtained by applying the imaginary time propagator $e^{-H \tau}$ to a given trial state $\psi_T$ for large imaginary time $\tau$ and sampling statistically the propagated state $ \psi_{\tau} = e^{-H \tau} \psi_T$. However the so called ``sign problem'' may appear in the simulation and such statistical propagation would be practically impossible without employing some approximation such as the well known ``fixed node'' approximation (FN). This method allows to improve the FN dynamic with a systematic correction scheme. This is possible by the simple requirement that, after a short imaginary time propagation via the FN dynamic, a number $p$ of correlation functions can be further constrained to be {\em exact} by small perturbation of the FN propagated state, which is free of the sign problem. By iterating this scheme the Monte Carlo average sign, which is almost zero when there is sign problem, remains stable and finite even for large $\tau$. The proposed algorithm is tested against the exact diagonalization results available on finite lattice. It is also shown in few test cases that the dependence of the results upon the few parameters entering the stochastic technique can be very easily controlled, unless for exceptional cases.