Encyclopedia of Algorithms

2016 Edition
| Editors: Ming-Yang Kao

Quantum Algorithms for Matrix Multiplication and Product Verification

  • Robin Kothari
  • Ashwin Nayak
Reference work entry
DOI: https://doi.org/10.1007/978-1-4939-2864-4_303

Years and Authors of Summarized Original Work

  • 2006; Buhrman, Špalek

  • 2012; Jeffery, Kothari, Magniez

Problem Definition

Let S be any algebraic structure over which matrix multiplication is defined, such as a field (e.g., real numbers), a ring (e.g., integers), or a semiring (e.g., the Boolean semiring). If we use + and ⋅ to denote the addition and multiplication operations over S, then the matrix product C of two n × n matrices A and B is defined as \(C_{ij} :=\sum \nolimits_{ k=1}^{n}A_{ik} \cdot B_{kj}\)

Keywords

Boolean matrix multiplication Matrix product verification Quantum algorithms 
This is a preview of subscription content, log in to check access.

Recommended Reading

  1. 1.
    Ambainis A (2007) Quantum walk algorithm for element distinctness. SIAM J Comput 37(1):210–239MathSciNetzbMATHCrossRefGoogle Scholar
  2. 2.
    Beals R, Buhrman H, Cleve R, Mosca M, de Wolf R (2001) Quantum lower bounds by polynomials. J ACM 48(4):778–797MathSciNetzbMATHCrossRefGoogle Scholar
  3. 3.
    Buhrman H, Špalek R (2006) Quantum verification of matrix products. In: Proceedings of 17th ACM-SIAM symposium on discrete algorithms, Miami, pp 880–889Google Scholar
  4. 4.
    Childs AM, Kimmel S, Kothari R (2012) The quantum query complexity of read-many formulas. In: Algorithms – ESA 2012. Volume 7501 of lecture notes in computer science. Springer, Heidelberg, pp 337–348Google Scholar
  5. 5.
    Farhi E, Goldstone J, Gutmann S, Sipser M (1998) Limit on the speed of quantum computation in determining parity. Phys Rev Lett 81(24):5442–5444CrossRefGoogle Scholar
  6. 6.
    Freivalds R (1979) Fast probabilistic algorithms. In: Mathematical foundations of computer science. Volume 74 of lecture notes in computer science. Springer, Berlin, pp 57–69Google Scholar
  7. 7.
    Grover LK (1996) A fast quantum mechanical algorithm for database search. In: Proceedings of the 28th ACM symposium on theory of computing (STOC 1996), Philadelphia, pp 212–219Google Scholar
  8. 8.
    Jeffery S, Kothari R, Magniez F (2012) Improving quantum query complexity of boolean matrix multiplication using graph collision. In: Automata, languages, and programming. Volume 7391 of lecture notes in computer science. Springer, Berlin/Heidelberg, pp 522–532Google Scholar
  9. 9.
    Kothari R (2014) An optimal quantum algorithm for the oracle identification problem. In: Proceedings of the 31st international symposium on theoretical aspects of computer science (STACS 2014), Lyon. Volume 25 of Leibniz international proceedings in informatics (LIPIcs), pp 482–493Google Scholar
  10. 10.
    Kothari R (2014) Efficient algorithms in quantum query complexity. PhD thesis, University of WaterlooGoogle Scholar
  11. 11.
    Le Gall F (2012) Improved output-sensitive quantum algorithms for Boolean matrix multiplication. In: Proceedings of the 23rd ACM-SIAM symposium on discrete algorithms (SODA 2012), Kyoto, pp 1464–1476Google Scholar
  12. 12.
    Le Gall F (2012) A time-efficient output-sensitive quantum algorithm for Boolean matrix multiplication. In: Algorithms and computation. Volume 7676 of lecture notes in computer science. Springer, Berlin, pp 639–648Google Scholar
  13. 13.
    Le Gall F, Nishimura H (2014) Quantum algorithms for matrix products over semirings. In: Algorithm theory – SWAT 2014. Volume 8503 of lecture notes in computer science. Springer, Berlin, pp 331–343Google Scholar
  14. 14.
    Magniez F, Nayak A, Roland J, Santha M (2011) Search via quantum walk. SIAM J Comput 40(1):142–164MathSciNetzbMATHCrossRefGoogle Scholar
  15. 15.
    Motwani R, Raghavan P (1995) Randomized algorithms. Cambridge University Press, New YorkzbMATHCrossRefGoogle Scholar
  16. 16.
    Santha M (2008) Quantum walk based search algorithms. In: Theory and applications of models of computation. Volume 4978 of lecture notes in computer science. Springer, New York, pp 31–46Google Scholar
  17. 17.
    Strassen V (1969) Gaussian elimination is not optimal. Numerische Mathematik 13:354–356MathSciNetzbMATHCrossRefGoogle Scholar
  18. 18.
    Williams VV, Williams R (2010) Subcubic equivalences between path, matrix and triangle problems. In: Proceedings of the 51st IEEE symposium on foundations of computer science (FOCS 2010), Las Vegas, pp 645–654Google Scholar

Copyright information

© Springer Science+Business Media New York 2016

Authors and Affiliations

  1. 1.Center for Theoretical PhysicsMassachusetts Institute of TechnologyCambridgeUSA
  2. 2.David R. Cheriton School of Computer ScienceInstitute for Quantum Computing, University of WaterlooWaterlooCanada
  3. 3.Department of Combinatorics and Optimization, Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada