Delayed Slater determinant update algorithms for high efficiency quantum Monte Carlo

TL;DR

This paper introduces a delayed update algorithm for Slater determinants in quantum Monte Carlo, significantly improving computational efficiency for large systems by applying multiple accepted moves simultaneously.

Contribution

A novel multiple rank delayed update scheme that enhances efficiency by batching matrix updates, applicable to large systems on CPUs and GPUs.

Findings

Order of magnitude speedup in update times for large systems

Effective on multi-core CPUs and GPUs

No change to Monte Carlo sampling or statistical properties

Abstract

Within ab initio Quantum Monte Carlo simulations, the leading numerical cost for large systems is the computation of the values of the Slater determinants in the trial wavefunction. Each Monte Carlo step requires finding the determinant of a dense matrix. This is most commonly iteratively evaluated using a rank-1 Sherman-Morrison updating scheme to avoid repeated explicit calculation of the inverse. The overall computational cost is therefore formally cubic in the number of electrons or matrix size. To improve the numerical efficiency of this procedure, we propose a novel multiple rank delayed update scheme. This strategy enables probability evaluation with application of accepted moves to the matrices delayed until after a predetermined number of moves, K. The accepted events are then applied to the matrices en bloc with enhanced arithmetic intensity and computational efficiency via matrix-matrix operations instead of matrix-vector operations. This procedure does not change the underlying Monte Carlo sampling or its statistical efficiency. For calculations on large systems and algorithms such as diffusion Monte Carlo where the acceptance ratio is high, order of magnitude improvements in the update time can be obtained on both multi-core CPUs and GPUs.