Determining QMC simulability with geometric phases

TL;DR

This paper establishes a geometric phase-based criterion for determining when non-stoquastic Hamiltonians can be efficiently simulated using quantum Monte Carlo methods, expanding the class of models accessible to such simulations.

Contribution

It provides a necessary and sufficient condition for QMC-simulability based on geometric phases, and constructs non-stoquastic yet QMC-simulable models.

Findings

Geometric phases characterize QMC-simulability of Hamiltonians.

Non-stoquastic models can be sign-problem-free under certain conditions.

QMC simulation of sign-problematic models can be optimized using alternative weights.

Abstract

Although stoquastic Hamiltonians are known to be simulable via sign-problem-free quantum Monte Carlo (QMC) techniques, the non-stoquasticity of a Hamiltonian does not necessarily imply the existence of a QMC sign problem. We give a sufficient and necessary condition for the QMC-simulability of Hamiltonians in a fixed basis in terms of geometric phases associated with the chordless cycles of the weighted graphs whose adjacency matrices are the Hamiltonians. We use our findings to provide a construction for non-stoquastic, yet sign-problem-free and hence QMC-simulable, quantum many-body models. We also demonstrate why the simulation of truly sign-problematic models using the QMC weights of the stoquasticized Hamiltonian is generally sub-optimal. We offer a superior alternative.