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2006, Quantum verification of matrix products

Harry Buhrman, Robert Špalek

Published Sep 6, 2004• 99 citations
OpenAlex

Abstract

We present a quantum algorithm that verifies a product of two n*n matrices over any field with bounded error in worst-case time n^{5/3} and expected time n^{5/3} / min(w,sqrt(n))^{1/3}, where w is the number of wrong entries. This improves the previous best algorithm that runs in time n^{7/4}. We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries.

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Improved Output-Sensitive Quantum Algorithms for Boolean Matrix Multiplication

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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.

ACM-SIAM Symposium on Discrete Algorithms (SODA)•Jan 17, 2012•17 citations
We present a quantum algorithm that verifies a product of two n*n matrices over any field with bounded error in worst-case time n^{5/3} and expected time n^{5/3} / min(w,sqrt(n))^{1/3}, where w is the number of wrong entries. This improves the previous best algorithm that runs in time n^{7/4}. We also present a quantum matrix multiplication algorithm that is efficient when the result has few nonzero entries.
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 propose an efficient quantum subroutine for matrix multiplication that computes a state vector encoding the entries of the product of two matrices in superposition. The subroutine exploits efficient state preparation techniques and shows a potential speed-up with respect to classical methods. The most important benefit of our subroutine is that it encodes the entries of the matrix product directly in the state vector, which can be used for further computations within the same quantum circuit. All scenarios involving the computation of non-homomorphic functions of the product of two matrices can benefit from our technique. As a possible application, we discuss the computation of the variance of the entries of a matrix product, which can be a useful tool for some machine learning algorithms.
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.