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arXiv · 2510.08509

Randomized and quantum approximate matrix multiplication

Simon Apers·Arjan Cornelissen·Samson Wang
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Abstract

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).

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Simon Apers, Arjan Cornelissen, Samson Wang. 2025-10-09. Randomized and quantum approximate matrix multiplication. https://arxiv.org/abs/2510.08509

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