Convergence Analysis of the Stochastic Resolution of Identity: Comparing Hutchinson to Hutch++ for the Second-Order Green's Function

TL;DR

This paper introduces a Hutch++ inspired stochastic resolution of identity method that significantly reduces statistical errors and improves computational efficiency in quantum chemistry calculations, especially for large systems.

Contribution

It develops a two-step stochastic resolution of identity approach employing randomized low-rank approximation, enhancing accuracy and efficiency over existing methods.

Findings

Hutch++-like method reduces statistical errors more effectively.

The approach outperforms deterministic and Hutchinson methods for large systems.

Significant efficiency gains at low error thresholds and intermediate system sizes.

Abstract

Stochastic orbital techniques offer reduced computational scaling and memory requirements to describe ground and excited states at the cost of introducing controlled statistical errors. Such techniques often rely on two basic operations, stochastic trace estimation and stochastic resolution of identity, both of which lead to statistical errors that scale with the number of stochastic realizations ($N_{\xi}$) as $\sqrt{N_{\xi}^{-1}}$. Reducing the statistical errors without significantly increasing $N_{\xi}$ has been challenging and is central to the development of efficient and accurate stochastic algorithms. In this work, we build upon recent progress made to improve stochastic trace estimation based on the ubiquitous Hutchinson's algorithm and propose a two-step approach for the stochastic resolution of identity, in the spirit of the Hutch++ method. Our approach is based on employing a randomized low-rank approximation followed by a residual calculation, resulting in statistical errors that scale much better than $\sqrt{N_{\xi}^{-1}}$. We implement the approach within the second-order Born approximation for the self-energy in the computation of neutral excitations and discuss three different low-rank approximations for the two-body Coulomb integrals. Tests on a series of hydrogen dimer chains with varying lengths demonstrate that the Hutch++-like approximations are computationally more efficient than both deterministic and purely stochastic (Hutchinson) approaches for low error thresholds and intermediate system sizes. Notably, for arbitrarily large systems, the Hutchinson-like approximation outperforms both deterministic and Hutch++-like methods.