Quantum Algorithms for Matrix Products over Semirings

In this paper we construct quantum algorithms for matrix products over several algebraic structures called semirings, including the (max,min)-matrix product, the distance matrix product and the Boolean matrix product. In particular, we obtain the following results. We construct a quantum algorithm computing the product of two n x n matrices over the (max,min) semiring with time complexity O(n2.473). In comparison, the best known classical algorithm for the same problem, by Duan and Pettie, has complexity O(n2.687). As an application, we obtain a O(n2.473)-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is O(n2.687), again by Duan and Pettie. We construct a quantum algorithm computing the L most significant bits of each entry of the distance product of two n x n matrices in time O(20.64L n2.46). In comparison, prior to the present work, the best known classical algorithm for the same problem, by Vassilevska and Williams and Yuster, had complexity O(2Ln2.69). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(20.96Ln2.69), which gives a sublinear dependency on 2L. The above two algorithms are the first quantum algorithms that perform better than the tilde O(n5/2)-time straightforward quantum algorithm based on quantum search for matrix multiplication over these semirings. We also consider the Boolean semiring, and construct a quantum algorithm computing the product of two n x n Boolean matrices that outperforms the best known classical algorithms for sparse matrices. For instance, if the input matrices have O(n1.686...) non-zero entries, then our algorithm has time complexity O(n2.277), while the best classical algorithm has complexity O(n2.373).