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 \(1\)s in the output, \(\ell \). We prove an upper bound of \(\widetilde{\hbox {O}}(n\sqrt{\ell +1})\) 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 \(\varepsilon < 1\) and any \(\ell \le \varepsilon n^2\), 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. Using similar ideas, we also show that the time complexity of Boolean matrix multiplication is \(\tilde{O}(n\sqrt{\ell +1}+\ell \sqrt{n})\).