Volume 2017

Quantum Algorithms for Matrix Products over Semirings

François Le Gall;
Kyoto University
Kyoto, Japan
legall AT i DOT kyoto-u DOT ac DOT jp

and

Harumichi Nishimura
Nagoya University
Nagoya, Japan
hnishimura AT math DOT cm DOT is DOT nagoya-u DOT ac DOT jp

May 11,2017

Abstract

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 $n\times n$ matrices over the (max,min) semiring with time complexity $O(n^{2.473})$. In comparison, the best known classical algorithm for the same problem, by Duan and Pettie (SODA'09), has complexity $O(n^{2.687})$.
• We construct a quantum algorithm computing the $\ell$ most significant bits of each entry of the distance product of two $n\times n$ matrices in time $O(2^{0.64\ell} n^{2.46})$. In comparison, the best known classical algorithm for the same problem, by Vassilevska and Williams (STOC'06) and Yuster (SODA'09), has complexity $O(2^{\ell}n^{2.69})$.
• The above two algorithms are the first quantum algorithms that perform better than the $\tilde O(n^{5/2})$-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 $n\times n$ Boolean matrices that outperforms the best known classical algorithms for sparse matrices.

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