Randomized and quantum approximate matrix multiplication

TL;DR

The paper tackles the problem of efficient approximate matrix multiplication by reframing randomized algorithms as mean-estimation tasks. It unifies classical random-walk and sketching approaches under a mean-estimation lens and then demonstrates quantum speedups via a multivariate quantum mean-estimation routine. Key contributions include a refined Cohen–Lewis classical algorithm with improved runtimes (when not using fast matrix multiplication), a unified sketching analysis for both Frobenius- and max-norm approximations, and quantum algorithms that achieve speedups in certain regimes, also generalizing to products of more than two matrices. The work outlines data-model trade-offs (RAM/QROM/QRAM/circuit) and norm-dependent performance, highlighting regimes where quantum advantages are unconditional and where classical sketching remains competitive, especially when leveraging fast matrix-multiplication techniques. Overall, the results illuminate how quantum mean-estimation techniques can accelerate approximate matrix multiplication and reveal nuanced interactions between norms, data models, and the use of fast matrix multiplication in practice.

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 Sarlós (`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).

Figures (2)

• Figure 1: Partial ordering of matrix norms appearing in Table \ref{['tab:results-introduction']}. Here, $\boldsymbol{X} , \boldsymbol{A} , \boldsymbol{B} \in \mathbb{R}^{n \times n}$, and $\boldsymbol{ \overline{ {A} } }$ represents the matrix containing all element-wise absolute values of $\boldsymbol{A}$. "$\rightarrow$" denotes "$\leq$". If there is no arrow in between two quantities then they are incomparable.
• Figure 2: A pictorial representation of the matrix product $\boldsymbol{A} _1 \cdots \boldsymbol{A} _k$ of stochastic matrices as a random walk on a directed graph. The random walk samples a vertex on the left side of the graph with probability distribution $\boldsymbol{d}$, and then transitions through the graph, with the probability transition matrix of every layer equal to $\boldsymbol{A} _j$. After $k$ steps, the probability distribution on the final layer of vertices is $\boldsymbol{d} ^T\boldsymbol{A} _1 \cdots \boldsymbol{A} _k$.

Theorems & Definitions (42)

• Definition 1.1: Approximate matrix multiplication
• Theorem 1.2: Simplified version of Theorems \ref{['thm:improved-cohen-lewis']} & \ref{['thm:quantum-cohen-lewis']}
• Theorem 1.3: Simplified version of Theorems \ref{['thm:Sarlos-multi']} and \ref{['thm:Sarlos-multi-fast']}
• Definition 2.1: Element-wise matrix norms
• Lemma 2.2: Hölder's inequality for element-wise matrix norms
• Lemma 2.3: Vector and matrix Bernstein inequality
• Theorem 2.4: Alias method vose1991linear
• Theorem 2.5: Quantum mean estimation cornelissen2022multiVariateMeanEstimation
• Theorem 3.1: Stochastic matrix product decomposition
• proof