Semistochastic Projector Monte Carlo Method

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.

💡 Research Summary

This paper presents a novel computational algorithm named the Semistochastic Projector Monte Carlo (SQMC) method, designed to efficiently solve for the dominant eigenvalue and eigenvector of extremely large matrices, a problem central to finding ground state properties in quantum many-body systems. The key innovation is a hybrid approach that partitions the matrix operation into two distinct regimes, combining the strengths of deterministic and stochastic techniques.

The method is built upon the power (or projector) method, which iteratively applies a transformed matrix P =