Improving Quantum Query Complexity of Boolean Matrix Multiplication Using Graph Collision

  • Stacey Jeffery
  • Robin Kothari
  • Frédéric Magniez
Part of the Lecture Notes in Computer Science book series (LNCS, volume 7391)

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 1s in the output, ℓ. We prove an upper bound of \(\widetilde{\mathrm{O}}(n\sqrt{\ell})\) for all values of ℓ. This is an improvement over previous algorithms for all values of ℓ. On the other hand, we show that for any ε < 1 and any ℓ ≤ εn2, 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2012

Authors and Affiliations

  • Stacey Jeffery
    • 1
    • 2
  • Robin Kothari
    • 1
    • 2
  • Frédéric Magniez
    • 3
  1. 1.David R. Cheriton School of Computer ScienceUniversity of WaterlooCanada
  2. 2.Institute for Quantum ComputingUniversity of WaterlooCanada
  3. 3.LIAFAUniv. Paris Diderot, CNRSParisFrance

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