Semistochastic Projector Monte Carlo Method

TL;DR

The paper presents a semistochastic power method that efficiently computes dominant eigenvalues and expectation values for large matrices, reducing computational time compared to fully stochastic methods, demonstrated on quantum systems with sign problems.

Contribution

It introduces a novel semistochastic implementation of the power method that combines exact and stochastic matrix multiplications, improving efficiency for large-scale quantum problems.

Findings

Significant reduction in computational time compared to fully stochastic methods

Successful application to fermion Hubbard model and carbon dimer

Effective handling of systems with sign problems

Abstract

We introduce a semistochastic implementation of the power method to compute, for very large matrices, the dominant eigenvalue and expectation values involving the corresponding eigenvector. The method is semistochastic in that the matrix multiplication is partially implemented numerically exactly and partially with respect to expectation values only. Compared to a fully stochastic method, the semistochastic approach significantly reduces the computational time required to obtain the eigenvalue to a specified statistical uncertainty. This is demonstrated by the application of the semistochastic quantum Monte Carlo method to systems with a sign problem: the fermion Hubbard model and the carbon dimer.