Determining QMC simulability with geometric phases

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. II. EMERGENCE OF THE SIGN PROBLEM AS A NONTRIVIAL GEOMETRIC PHASEWe now examine in detail the origins of the sign problem in QMC simulations. By doing so, we are able to differentiate QMC-simulable Hamiltonians (in a given basis) from non-simulable ones.