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.

翻译：Hutchinson估计器是一种随机算法，通过对任意半正定矩阵使用$\mathcal{O}(1/\epsilon^2)$次矩阵-向量乘积，计算其迹的$\epsilon$-近似。Hutchinson估计器的改进版本Hutch++仅需$\mathcal{O}(1/\epsilon)$次矩阵-向量乘积。本文提出Hutch++的推广形式，即ContHutch++，它利用算子-函数乘积高效估计任意迹类积分算子的迹。ContHutch++的估计避免了离散化引入的光谱伪影，并伴随严格的高概率误差界。我们利用ContHutch++推导了一种用于量子态密度的全新高阶精确算法，并展示了该算法如何估计非相干源激发的电磁场。