ContHutch++: Stochastic trace estimation for implicit integral operators

TL;DR

ContHutch++ is a novel stochastic trace estimation method for integral operators that improves accuracy and efficiency, with applications in quantum physics and electromagnetic field estimation.

Contribution

It generalizes Hutch++, using operator-function products for better trace estimation of integral operators with rigorous error bounds.

Findings

Provides high-probability error bounds for estimations.

Achieves efficient trace estimation avoiding spectral artifacts.

Demonstrates applications in quantum density-of-states and electromagnetic fields.

Abstract

Hutchinson's estimator is a randomized algorithm that computes an $\epsilon$-approximation to the trace of any positive semidefinite matrix using $\mathcal{O}(1/\epsilon^2)$ matrix-vector products. An improvement of Hutchinson's estimator, known as Hutch++, only requires $\mathcal{O}(1/\epsilon)$ matrix-vector products. In this paper, we propose a generalization of Hutch++, which we call ContHutch++, that uses operator-function products to efficiently estimate the trace of any trace-class integral operator. Our ContHutch++ estimates avoid spectral artifacts introduced by discretization and are accompanied by rigorous high-probability error bounds. We use ContHutch++ to derive a new high-order accurate algorithm for quantum density-of-states and also show how it can estimate electromagnetic fields induced by incoherent sources.