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

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