Improved output-sensitive quantum algorithms for Boolean matrix multiplication

We present new quantum algorithms for Boolean Matrix Multiplication in both the time complexity and the query complexity settings. As far as time complexity is concerned, our results show that the product of two n× n Boolean matrices can be computed on a quantum computer in time Õ(n3/2+nl3/4), where l is the number of non-zero entries in the product, improving over the outputsensitive quantum algorithm by Buhrman and Špalek (SODA’06) that runs in Õ(n3/2 √ l) time. This is done by constructing a quantum version of a recent classical algorithm by Lingas (ESA’09), using quantum techniques such as quantum counting to exploit the sparsity of the output matrix. As far as query complexity is concerned, our results improve over the quantum algorithm by Vassilevska Williams and Williams (FOCS’10) based on a reduction to the triangle finding problem. One of the main contributions leading to this improvement is the construction of a quantum algorithm for triangle finding tailored especially for the tripartite graphs appearing in the reduction. Boolean matrix multiplication, where addition is interpreted as a logical OR and multiplication as a logical AND, is a fundamental problem in computer science. Algorithms for Boolean matrix multiplication have found applications in many areas and are, for example, used to construct efficient algorithms for computing the transitive closure of a graph [7, 8, 15], recognizing context-free languages [16, 20], detecting if a graph contains a triangle [11], solving all-pairs path problems [6, 9, 18, 19], or speeding up data mining tasks [1]. The product of two Boolean n× n matrices A and B can be trivially computed in time O(n3). The best known classical algorithm is obtained by seeing the matrices A and B as integer matrices, computing the integer matrix product, and converting the product matrix to a Boolean matrix. Using the algorithm by Coppersmith and Winograd [5] for integer matrix multiplication, this gives an algorithm for Boolean matrix multiplication with time complexity O(n2.376). This approach has nevertheless several disadvantages, the main one being that Coppersmith-Winograd’s algorithm can be hard to implement in practice. Partly for this reason, much effort has focused on understanding whether Boolean matrix multiplication can be done in o(n3) time by combinatorial algorithms, i.e., classical algorithms that do not rely on a product of matrices over rings. A maybe more fundamental reason for investigating this question is that a fast combinatorial algorithm for matrix multiplication over the semi-ring (OR, AND) would possibly generalize to other semi-rings, and especially to semi-rings such as (min,+) related to a multitude of problems over weighted graphs such as the all-pairs shortest paths problem, over which no subcubic time multiplication algorithm is available. Unfortunately, there have been little progress on this question. The best known combinatorial algorithm has time complexity O(n3/ log2.25(n)) and has been discovered recently by Bansal and Williams [3], improving the “four Russians” algorithm [2] proposed decades ago. In the quantum computation model, there exist subcubic-time algorithms for Boolean matrix multiplication that do not rely on integer matrix multiplication. Indeed, the product of two n× n Boolean matrices A and B can be easily computed in time Õ(n2.5): for each pair of indexes i, j ∈ {1,2, . . . ,n}, check if there exists an index k ∈ {1, . . . ,n} such that A[i,k] = B[k, j] = 1 in time Õ( √ n) using Grover’s quantum search algorithm [10]. Buhrman and Špalek [4] observed that a similar argument can be used