Determining quantum Monte Carlo simulability with geometric phases

Itay Hen

doi:10.1103/physrevresearch.3.023080

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.

Highlights

The concept of stoquasticity [1] is a key definition in both quantum Monte Carlo (QMC) simulations and computational complexity theory
II, we present a generic partition function decomposition focusing on the signs of its summands and their origins, which we trace back to geometric phases associated with the graph structure of the Hamiltonian
We found that if and only if all the geometric phases of the chordless cycles of the weighted graph whose adjacency matrix is the Hamiltonian are zero, the simulation will be SPF, a condition we call vanishing geometric phase (VGP)

Summary

INTRODUCTION

The concept of stoquasticity [1] is a key definition in both quantum Monte Carlo (QMC) simulations and computational complexity theory. In the field of QMC simulations [5,6], the partition function of stoquastic Hamiltonians can always be written as a sum of efficiently computable strictly positive weights [7,8,9]. As a consequence, such Hamiltonians do not suffer from a sign problem [10,11], i.e., from the existence of negative summands, which greatly impede the convergence of QMC algorithms [10,11,12,13].

Permutation matrix representation of Hamiltonians

Partition function decomposition

From walks to cycles

NONSTOQUASTIC SIGN-PROBLEM-FREE HAMILTONIANS

QMC SIMULATIONS WITHOUT STOQUASTIZATION

SUMMARY AND CONCLUSIONS