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 × 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 ℓ most significant bits of each entry of the distance product of two n × n matrices in time O(2 0.64ℓ 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 ℓ n 2.69 ).The above two algorithms are the first quantum algorithms that perform better than the Õ(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 × n Boolean matrices that outperforms the best known classical algorithms for sparse matrices.
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 × 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 ℓ most significant bits of each entry of the distance product of two n × n matrices in time O(2 0.64ℓ 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 ℓ n 2.69 ).The above two algorithms are the first quantum algorithms that perform better than the Õ(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 × n Boolean matrices that outperforms the best known classical algorithms for sparse matrices.
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(n 3/2 + nℓ 3/4), where ℓ is the number of non-zero entries in the product, improving over the output-sensitive quantum algorithm by Buhrman and ˇ Spalek that runs in Õ(n3/2√ℓ) time. This is done by constructing a quantum version of a recent algorithm by Lingas, 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 based on a reduction to the triangle finding problem. One of the main contributions leading to this improvement is the construction of a triangle finding quantum algorithm tailored especially for the tripartite graphs appearing in the reduction. 1
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 output-sensitive quantum algorithm by Buhrman and Spalek that runs in Õ(n3/2 √ℓ) time. This is done by constructing a quantum version of a recent algorithm by Lingas, 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 based on a reduction to the triangle finding problem. One of the main contributions leading to this improvement is the construction of a triangle finding quantum algorithm tailored especially for the tripartite graphs appearing in the reduction.
The problem of finding maximum (or minimum) witnesses of the Boolean product of two Boolean matrices (MW for short) has a number of important applications, in particular the all-pairs lowest common ancestor (LCA) problem in directed acyclic graphs (dags). The best known upper time-bound on the MW problem for [Formula: see text] Boolean matrices of the form [Formula: see text] has not been substantially improved since 2006. In order to obtain faster algorithms for this problem, we study quantum algorithms for MW and approximation algorithms for MW (in the classical computational model). Some of our quantum algorithms are input or output sensitive. Our fastest quantum algorithm for the MW problem, and consequently for the related problems, runs in time [Formula: see text], where [Formula: see text] satisfies the equation [Formula: see text] and [Formula: see text] is the exponent of the multiplication of an [Formula: see text] matrix by an [Formula: see text] matrix. Next, we consider a relaxed version of the MW problem (in the classical model) asking for reporting a witness of bounded rank (the maximum witness has rank 1) for each non-zero entry of the matrix product. First, by adapting the fastest known algorithm for maximum witnesses, we obtain an algorithm for the relaxed problem that reports for each non-zero entry of the product matrix a witness of rank at most [Formula: see text] in time [Formula: see text] Then, by reducing the relaxed problem to the so called [Formula: see text]-witness problem, we provide an algorithm that reports for each non-zero entry [Formula: see text] of the product matrix [Formula: see text] a witness of rank [Formula: see text], where [Formula: see text] is the number of witnesses for [Formula: see text], with high probability. The algorithm runs in [Formula: see text] time, where [Formula: see text].
We prove the first non-trivial upper bound for the quantum query complexity of Boolean Matrix Product Verification (BMPV), answering a longstanding open question in quantum query complexity. For n×n matrices, our upper bound is O(n17/12), improving on the standard O(n3/2) bound obtained using Grover search by Buhrman and Špalek [SODA 2006]. We complement this result by showing an Ω(n5/4) lower bound, which improves over the previous best known lower bound of Ω(n19/18) by Childs, Kimmel, and Kothari [ESA 2012]. Our approach centers on a connection with Orthogonal Vectors (OV), which asks whether an indexed list of n Boolean vectors of dimension n contains two vectors with disjoint supports. In particular, we prove equivalences between OV and BMPV and establish the above bounds for OV. We also prove a tight Θ(n3/2) bound for a variant of BMPV that asks whether the product contains a given row vector. Together, these results imply a polynomial separation between the quantum query complexities of deciding whether a graph has radius at most two and whether it has diameter at most two.