Advertisement

Quantum Algorithms for Matrix Products over Semirings

  • François Le Gall
  • Harumichi Nishimura
Part of the Lecture Notes in Computer Science book series (LNCS, volume 8503)

Abstract

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 complexity O(n 2.687). As an application, we obtain a O(n 2.473)-time quantum algorithm for computing the all-pairs bottleneck paths of a graph with n vertices, while classically the best upper bound for this task is 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(20.64ℓ n 2.46). In comparison, prior to the present work, the best known classical algorithm for the same problem had complexity O(2 n 2.69). Our techniques lead to further improvements for classical algorithms as well, reducing the classical complexity to O(20.96ℓ n 2.69), which gives a sublinear dependency on 2.

The above two algorithms are the first quantum algorithms that perform better than the \(\tilde O(n^{5/2})\)-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.

Keywords

Matrix Multiplication Matrix Product Quantum Algorithm Classical Algorithm Boolean Matrix 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Amossen, R.R., Pagh, R.: Faster join-projects and sparse matrix multiplications. In: Proceedings of ICDT, pp. 121–126 (2009)Google Scholar
  2. 2.
    Boyer, M., Brassard, G., Høyer, P., Tapp, A.: Tight bounds on quantum searching. Fortschritte der Physik 46(4-5), 493–505 (1998)CrossRefGoogle Scholar
  3. 3.
    Buhrman, H., Špalek, R.: Quantum verification of matrix products. In: Proceedings of SODA, pp. 880–889 (2006)Google Scholar
  4. 4.
    Bürgisser, P., Clausen, M., Shokrollahi, M.A.: Algebraic complexity theory. Springer (1997)Google Scholar
  5. 5.
    Duan, R., Pettie, S.: Fast algorithms for (max, min)-matrix multiplication and bottleneck shortest paths. In: Proceedings of SODA, pp. 384–391 (2009)Google Scholar
  6. 6.
    Dubois, D., Prade, H.: Fuzzy sets and systems: Theory and applications. Academic Press (1980)Google Scholar
  7. 7.
    Dürr, C., Høyer, P.: A quantum algorithm for finding the minimum. arXiv:quant-ph/9607014 (1996)Google Scholar
  8. 8.
    Grover, L.K.: A fast quantum mechanical algorithm for database search. In: Proceedings of STOC, pp. 212–219 (1996)Google Scholar
  9. 9.
    Huang, X., Pan, V.Y.: Fast rectangular matrix multiplication and applications. Journal of Complexity 14(2), 257–299 (1998)CrossRefzbMATHMathSciNetGoogle Scholar
  10. 10.
    Jeffery, S., Kothari, R., Magniez, F.: Improving quantum query complexity of Boolean matrix multiplication using graph collision. In: Czumaj, A., Mehlhorn, K., Pitts, A., Wattenhofer, R. (eds.) ICALP 2012, Part I. LNCS, vol. 7391, pp. 522–532. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  11. 11.
    Le Gall, F.: Faster algorithms for rectangular matrix multiplication. In: Proceedings of FOCS, pp. 514–523 (2012)Google Scholar
  12. 12.
    Le Gall, F.: Improved output-sensitive quantum algorithms for Boolean matrix multiplication. In: Proceedings of SODA, pp. 1464–1476 (2012)Google Scholar
  13. 13.
    Le Gall, F.: A time-efficient output-sensitive quantum algorithm for Boolean matrix multiplication. In: Chao, K.-M., Hsu, T.-S., Lee, D.-T. (eds.) ISAAC 2012. LNCS, vol. 7676, pp. 639–648. Springer, Heidelberg (2012)CrossRefGoogle Scholar
  14. 14.
    Le Gall, F.: Powers of tensors and fast matrix multiplication. In: Proceedings of ISSAC (to appear, 2014)Google Scholar
  15. 15.
    Le Gall, F., Nishimura, H.: Quantum algorithms for matrix products over semirings. Full version of the present paper, available as arXiv:1310.3898Google Scholar
  16. 16.
    Magniez, F., Santha, M., Szegedy, M.: Quantum algorithms for the triangle problem. SIAM Journal on Computing 37(2), 413–424 (2007)CrossRefzbMATHMathSciNetGoogle Scholar
  17. 17.
    Matoušek, J.: Computing dominances in E n. Information Processing Letters 38(5), 277–278 (1991)CrossRefzbMATHMathSciNetGoogle Scholar
  18. 18.
    Shapira, A., Yuster, R., Zwick, U.: All-pairs bottleneck paths in vertex weighted graphs. In: Proceedings of SODA, pp. 978–985 (2007)Google Scholar
  19. 19.
    Vassilevska, V.: Efficient Algorithms for Path Problems in Weighted Graphs. PhD thesis, Carnegie Mellon University (2008)Google Scholar
  20. 20.
    Vassilevska, V., Williams, R.: Finding a maximum weight triangle in n 3 − δ time, with applications. In: Proceedings of STOC, pp. 225–231 (2006)Google Scholar
  21. 21.
    Vassilevska, V., Williams, R., Yuster, R.: All pairs bottleneck paths and max-min matrix products in truly subcubic time. Theory of Computing 5(1), 173–189 (2009)CrossRefMathSciNetGoogle Scholar
  22. 22.
    Vassilevska Williams, V.: Multiplying matrices faster than Coppersmith-Winograd. In: Proceedings of STOC, pp. 887–898 (2012)Google Scholar
  23. 23.
    Yuster, R.: Efficient algorithms on sets of permutations, dominance, and real-weighted APSP. In: Proceedings of SODA, pp. 950–957 (2009)Google Scholar
  24. 24.
    Yuster, R., Zwick, U.: Fast sparse matrix multiplication. ACM Transactions on Algorithms 1(1), 2–13 (2005)CrossRefMathSciNetGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  • François Le Gall
    • 1
  • Harumichi Nishimura
    • 2
  1. 1.Graduate School of Information Science and TechnologyThe University of TokyoJapan
  2. 2.Graduate School of Information ScienceNagoya UniversityJapan

Personalised recommendations