Randomized and quantum approximate matrix multiplication

2025-10-09

AI Summary

Approximate matrix multiplication is a fundamental problem in theoretical computer science. This paper unifies classical randomized algorithms—including random walks and sketching—under a mean estimation framework, yielding a novel classical approximation algorithm that avoids invoking exact matrix multiplication as a subroutine and achieves improved time complexity over prior methods. Furthermore, leveraging quantum multivariate mean estimation, we achieve the first quantum speedup for approximate matrix multiplication without relying on quantum fast matrix multiplication, thereby breaking classical lower bounds. Both theoretical analysis and empirical evaluation demonstrate significant efficiency gains on sparse and low-rank structured matrices. The proposed framework establishes a new paradigm for large-scale approximate matrix computation, balancing theoretical rigor with practical applicability.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Search and Optimization: Non-convex Optimization
• Constraint Satisfaction and Optimization: Distributed CSP/Optimization

Application Category

• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

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).

Problem

• Refining classical randomized matrix multiplication algorithms
• Developing improved classical algorithm without fast matrix multiplication
• Demonstrating quantum speedup using quantum mean estimation

Innovation

• Refined analysis of classical randomized matrix algorithms
• Improved classical algorithm outperforming prior approaches
• Quantum speedup via multivariate mean estimation technique