Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

TL;DR

This paper presents a quantum algorithm that improves the query complexity for Boolean matrix multiplication by reducing it to graph collision problems and exploiting graph density, achieving near-optimal bounds.

Contribution

The paper introduces a new quantum algorithm with an upper bound of O(n√ℓ) for Boolean matrix multiplication, matching a proven lower bound for dense cases.

Findings

Achieves an upper bound of O(n√ℓ) for all ℓ values.

Establishes an Ω(n√ℓ) lower bound for dense matrices, showing near-optimality.

Reduces matrix multiplication to graph collision problems for efficient quantum solutions.

Abstract

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, n, as well as the number of 1s in the output, \ell. We prove an upper bound of O (n\sqrt{\ell}) for all values of \ell. This is an improvement over previous algorithms for all values of \ell. On the other hand, we show that for any \eps < 1 and any \ell <= \eps n^2, there is an \Omega(n\sqrt{\ell}) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently.