A Time-Efficient Output-Sensitive Quantum Algorithm for Boolean Matrix Multiplication

TL;DR

This paper introduces a quantum algorithm for Boolean matrix multiplication that is output-sensitive and more efficient than previous algorithms, with proven limits on potential improvements.

Contribution

The paper presents a novel quantum algorithm for Boolean matrix multiplication with improved output-sensitive time complexity and establishes fundamental limits on further enhancements.

Findings

Achieves $ ilde O(n\sqrt{ ext{ell}}+ ext{ell}\sqrt{n})$ time complexity

Improves upon previous quantum algorithms by Buhrman and Špalek, Le Gall

Proves that further improvements would imply a breakthrough in quantum matrix multiplication

Abstract

This paper presents a quantum algorithm that computes the product of two $n\times n$ Boolean matrices in $\tilde O(n\sqrt{\ell}+\ell\sqrt{n})$ time, where $\ell$ is the number of non-zero entries in the product. This improves the previous output-sensitive quantum algorithms for Boolean matrix multiplication in the time complexity setting by Buhrman and \v{S}palek (SODA'06) and Le Gall (SODA'12). We also show that our approach cannot be further improved unless a breakthrough is made: we prove that any significant improvement would imply the existence of an algorithm based on quantum search that multiplies two $n\times n$ Boolean matrices in $O(n^{5/2-\varepsilon})$ time, for some constant $\varepsilon>0$.