Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

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 Ω(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.

References in corpus (1)

• The quantum query complexity of read-many formulas

Cited by in corpus (6)

• Learning-Graph-Based Quantum Algorithm for k-distinctness
• A quantum query algorithm for the graph collision problem
• Applications of the Adversary Method in Quantum Query Algorithms
• Parameterized Quantum Query Complexity of Graph Collision
• On Nondeterministic Derandomization of Freivalds' Algorithm: Consequences, Avenues and Algorithmic Progress
• Quantum Algorithms for Matrix Products over Semirings