Improving quantum query complexity of boolean matrix multiplication using graph collision

Authors:
Stacey Jeffery;Robin Kothari;Frédéric Magniez
Affiliations:
David R. Cheriton School of Computer Science, University of Waterloo, Canada, Institute for Quantum Computing, University of Waterloo, Canada;David R. Cheriton School of Computer Science, University of Waterloo, Canada, Institute for Quantum Computing, University of Waterloo, Canada;LIAFA, Univ. Paris Diderot, CNRS, Paris, France
Venue:
ICALP'12 Proceedings of the 39th international colloquium conference on Automata, Languages, and Programming - Volume Part I
Year:
2012

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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 εεn2, 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.