Rapid mixing of path integral Monte Carlo for 1D stoquastic Hamiltonians

Path integral quantum Monte Carlo (PIMC) is a method for estimating thermal equilibrium properties of stoquastic spin systems by sampling from classical Gibbs distribution using Markov chain Carlo. The PIMC has been widely used to study the physics materials and simulated annealing, but these successful applications are rarely accompanied formal proofs that chains underlying rapidly converge desired distribution. In this work we analyze mixing time 1D Hamiltonians, including disordered transverse Ising models (TIM) with long-range algebraically decaying interactions as well XY nearest-neighbor interactions. By bounding convergence rigorously justify use approximate partition functions expectations observables at inverse temperatures scale most logarithmically number qubits. analysis based on canonical paths applied single-site Metropolis 2D couplings related in Hamiltonian. Since system strongly nonisotropic grow size, it does not fall into known cases where mix rapidly.