Fast mixing of operator-loop path-integral quantum Monte Carlo for stoquastic XY Hamiltonians

Quantum Monte Carlo method with operator-loop update is a powerful technique that has been extensively used with great success in condensed matter physics. It enables one to sample from thermal and ground states of local Hamiltonians of various spin, bosonic and fermionic systems as long as the Hamiltonian does not have a negative-sign problem. Despite the practical success of this method, theoretical understanding of the efficiency of the algorithm has been lacking. The operator-loop update is commonly used for path-integral formulation (Suzuki-Trotter/world-lines) of the partition function. In this work we consider this method applied to the stoquastic (sign-problem free) XY model and prove that the mixing time of the Markov chain is polynomial in the system size and the inverse temperature. Using the fast mixing Markov chain, we can estimate the partition functions of the Hamiltonians that we consider in a polynomial time, significantly improving upon the best known previous algorithm by Bravyi and Gosset [arXiv:1612.05602]. Our algorithm also allows for natural extensions to a wide class of empirically fast-mixing Hamiltonians.