The complexity of matrix multiplication is a central topic in computer science. While the focus has traditionally been on exact algorithms, a long line of literature also considers randomized algorithms, which return an approximate solution in faster time. In this work, we adopt a unifying perspective that frames these randomized algorithms in terms of mean estimation. Using it, we first give refined analyses of classical algorithms based on random walks by Cohen-Lewis (`99), and based on sketching by Sarl\'os (`06) and Drineas-Kannan-Mahoney (`06). We then propose an improvement on Cohen-Lewis that yields a single classical algorithm that is faster than all the other approaches, if we assume no use of (exact) fast matrix multiplication as a subroutine. Second, we demonstrate a quantum speedup on top of these algorithms by using the recent quantum multivariate mean estimation algorithm by Cornelissen-Hamoudi-Jerbi (`22).

翻译：矩阵乘法的复杂度是计算机科学的核心课题。传统研究主要关注精确算法，但长期以来也有大量文献研究随机化算法，这类算法能以更快的速度返回近似解。本文采用统一视角，将随机化算法框架化为均值估计问题。基于此框架，我们首先对基于随机游走的经典算法（Cohen-Lewis, `99）以及基于素描技术的经典算法（Sarlós, `06; Drineas-Kannan-Mahoney, `06）进行了精细化分析。随后，我们在Cohen-Lewis算法基础上提出改进方案，该方案在不使用（精确）快速矩阵乘法作为子程序的前提下，构建出单一经典算法，其速度优于所有其他方法。其次，我们通过应用Cornelissen-Hamoudi-Jerbi (`22) 提出的量子多元均值估计算法，在上述算法基础上实现了量子加速。