Systematic lowering of the scaling of Monte Carlo calculations by partitioning andsubsampling

TL;DR

This paper introduces a partitioning and subsampling approach in Monte Carlo calculations that reduces the scaling of statistical fluctuations, significantly improving efficiency for extensive properties in many-body systems.

Contribution

The authors develop a novel Monte Carlo method using space partitioning and subsampling that achieves linear scaling in computational cost for extensive observables.

Findings

Achieves O(N) scaling in numerical efficiency.

Demonstrates effectiveness on 2D Hubbard model and metallic hydrogen chains.

No bias introduced, with a zero-variance limit in separable cases.

Abstract

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 2D Hubbard model and on metallic hydrogen chains using Jastrow-Slater wave functions. A factor O(N) is gained in numerical efficiency.