# Quantum Algorithms for Matrix Products over Semirings

@inproceedings{Gall2017QuantumAF,
title={Quantum Algorithms for Matrix Products over Semirings},
author={François Le Gall and Harumichi Nishimura},
booktitle={Chic. J. Theor. Comput. Sci.},
year={2017}
}
• Published in Chic. J. Theor. Comput. Sci. 15 October 2013
• Computer Science, Mathematics
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 has…

## Figures from this paper

Quantum hyperparallel algorithm for matrix multiplication
• Computer Science, Physics
Scientific reports
• 2016
A hyperparallel quantum algorithm for matrix multiplication with time complexity O(N2), which is better than the best known classical algorithm and shows that hyperpar parallel quantum computation may provide a useful tool in quantum machine learning and “big data” analysis.
Quantum Algorithms for Matrix Multiplication and Product Verification
• Computer Science
Encyclopedia of Algorithms
• 2016
This work has shown that quantum algorithms for matrix multiplication and product verification can be implemented as discrete-time solutions to the challenge of solving matrix multiplicationand product verification problems.
Approximating APSP without scaling: equivalence of approximate min-plus and exact min-max
• Computer Science
STOC
• 2019
It is proved that approximating directed APSP and exactly computing the Min-Max Product are equivalent, and the first strongly polynomial approximation scheme for Min-Plus Convolution in strongly subquadratic time is obtained and an equivalence of approximate Min- plus Convolution and exact Min- Max Convolution is proved.
Faster Algorithms for All Pairs Non-decreasing Paths Problem
• Computer Science, Mathematics
ICALP
• 2019
An improved algorithm for the All Pairs Non-decreasing Paths (APNP) problem on weighted simple digraphs is presented, which has running time $\tilde{O}(n^2)$ time algorithm for APNP in undirected graphs which also reaches optimal within logarithmic factors.
Quantum Algorithm Design: Techniques and Applications
• Computer Science
J. Syst. Sci. Complex.
• 2019
An overview of the development of quantum algorithms, then five important techniques are investigated: Quantum phase estimation, linear combination of unitaries, quantum linear solver, Grover search, and quantum walk, together with their applications in quantum state preparation, quantum machine learning, and Quantum search.

## References

SHOWING 1-10 OF 30 REFERENCES
Improved output-sensitive quantum algorithms for Boolean matrix multiplication
The results show that the product of two n x n Boolean matrices can be computed on a quantum computer in time O(n3/2 + nl3/4), improving over the output-sensitive quantum algorithm by Buhrman and Spalek that runs in [EQUATION] time.
A Time-Efficient Output-Sensitive Quantum Algorithm for Boolean Matrix Multiplication
It is proved that any significant improvement would imply the existence of an algorithm based on quantum search that multiplies two Boolean matrices in O(n^{5/2-\varepsilon}) time, for some constant $\varePSilon>0$.
A Fast Output-Sensitive Algorithm for Boolean Matrix Multiplication
This work uses randomness to exploit the potential sparsity of the Boolean matrix product in order to speed up the computation of the product and presents a new fast output-sensitive algorithm that provides the Boolean product and its witnesses almost certainly.
On the complexity of matrix multiplication
The evaluation of the product of two matrices can be very computationally expensive. The multiplication of two n×n matrices, using the “default” algorithm can take O(n3) field operations in the
Tight bounds on quantum searching
• Computer Science
• 1996
A lower bound on the efficiency of any possible quantum database searching algorithm is provided and it is shown that Grover''s algorithm nearly comes within a factor 2 of being optimal in terms of the number of probes required in the table.
A Quantum Algorithm for Finding the Minimum
• Computer Science
ArXiv
• 1996
A simple quantum algorithm whicholves the minimum searching problem using O(√N) probes using the mainsubroutine of Grover’s recent quantum searching algorithm.
Faster Algorithms for Rectangular Matrix Multiplication
• F. Gall
• Computer Science, Mathematics
2012 IEEE 53rd Annual Symposium on Foundations of Computer Science
• 2012
A new algorithm for multiplying an n × n<sup>k</sup> matrix by an n–k × n matrix, which is better than all known algorithms for rectangular matrix multiplication and recovers exactly the complexity of the algorithm by Coppersmith and Winograd.
Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths
• Computer Science
SODA
• 2009
This paper gives an APBSP algorithm for edge-capacitated graphs running in O(n(3+ω)/2) time and a slightly faster O( n2.657)-time algorithm for vertex-capactitated graphs, and makes use of new hybrid products the authors call the distance-max-min product and dominance-distance product.
Fast Rectangular Matrix Multiplication and Applications
• Mathematics, Computer Science
J. Complex.
• 1998
This work presents fast multiplication algorithms for matrix pairs of arbitrary dimensions, estimates the asymptotic running time as a function of the dimensions, and improves the exponents of the complexity estimates for computing basic solutions to the linear programming problem with constraints andvariables.
Efficient algorithms on sets of permutations, dominance, and real-weighted APSP
Two algorithms, each solving a different problem, use fast matrix multiplication techniques to achieve a significant improvement in the running time over the naive solutions, and present a randomized algorithm that decides if P is fully expanding.