Efficient Sampling of Thermal Averages of Interacting Quantum Particle Systems with Random Batches

An efficient sampling method, the pmmLang+RBM, is proposed to compute the quantum thermal average in the interacting quantum particle system. Benefiting from the random batch method (RBM), the pmmLang+RBM reduces the complexity due to the interaction forces per timestep from $O(NP^2)$ to $O(NP)$, where $N$ is the number of beads and $P$ is the number of particles. Although the RBM introduces a random perturbation of the interaction forces at each timestep, the long time effects of the random perturbations along the sampling process only result in a small bias in the empirical measure of the pmmLang+RBM from the target distribution, which also implies a small error in the thermal average calculation. We numerically study the convergence of the pmmLang+RBM, and quantitatively investigate the dependence of the error in computing the thermal average on the parameters including the batch size, the timestep, etc. We also propose an extension of the pmmLang+RBM, which is based on the splitting Monte Carlo method and is applicable when the interacting potential contains a singular part.