Systematic lowering of the scaling of Monte Carlo calculations by partitioning and subsampling

We propose to compute physical properties by Monte Carlo calculations using conditional expectation values. The latter are obtained on top of the usual Monte Carlo sampling by partitioning the physical space in several subspaces or fragments, and subsampling each fragment (i.e., performing side walks) while freezing the environment. No bias is introduced and a zero-variance principle holds in the limit of separability, i.e., when the fragments are independent. In practice, the usual bottleneck of Monte Carlo calculations???the scaling of the statistical fluctuations as a function of the number of particles N???is relieved for extensive observables. We illustrate the method in variational Monte Carlo on the two-dimensional Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.

Automated summary

This study proposes a method to calculate physical properties using Monte Carlo calculations and conditional expectation values. By partitioning the physical space into subspaces or fragments and subsampling each fragment while freezing the environment, the conditional expectation values are obtained without introducing bias, and a zero-variance principle is maintained in the limit of separability. The method alleviates the usual bottleneck of Monte Carlo calculations in terms of scaling statistical fluctuations with the number of particles N, particularly for extensive observables.