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Algorithmica

, Volume 76, Issue 1, pp 1–16 | Cite as

Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

  • Stacey Jeffery
  • Robin Kothari
  • François Le Gall
  • Frédéric Magniez
Article

Abstract

The quantum query complexity of Boolean matrix multiplication is typically studied as a function of the matrix dimension, \(n\), as well as the number of \(1\)s in the output, \(\ell \). We prove an upper bound of \(\widetilde{\hbox {O}}(n\sqrt{\ell +1})\) for all values of \(\ell \). This is an improvement over previous algorithms for all values of \(\ell \). On the other hand, we show that for any \(\varepsilon < 1\) and any \(\ell \le \varepsilon n^2\), there is an \(\Omega (n\sqrt{\ell })\) lower bound for this problem, showing that our algorithm is essentially tight. We first reduce Boolean matrix multiplication to several instances of graph collision. We then provide an algorithm that takes advantage of the fact that the underlying graph in all of our instances is very dense to find all graph collisions efficiently. Using similar ideas, we also show that the time complexity of Boolean matrix multiplication is \(\tilde{O}(n\sqrt{\ell +1}+\ell \sqrt{n})\).

Keywords

Quantum algorithms Boolean matrix multiplication Query complexity 

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Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  • Stacey Jeffery
    • 1
    • 2
  • Robin Kothari
    • 1
    • 2
  • François Le Gall
    • 3
  • Frédéric Magniez
    • 4
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooWaterlooCanada
  2. 2.Institute for Quantum ComputingUniversity of WaterlooWaterlooCanada
  3. 3.Graduate School of Information Science and TechnologyThe University of TokyoTokyoJapan
  4. 4.Univ Paris Diderot, Sorbonne Paris-CitéParisFrance

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