Quantum Complexity of Testing Group Commutativity

Authors:
Frederic Magniez;Ashwin Nayak
Affiliations:
CNRS, LRI - batiment 490, Universite Paris-Sud, 91405 Orsay cedex, France;Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo, 200 University Ave. W., Waterloo, Ontario N2L 3G1, Canada
Venue:
Algorithmica
Year:
2007

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Abstract

We consider the problem of testing the commutativity of a black-box group specified by its k generators. The complexity (in terms of k) of this problem was first considered by Pak, who gave a randomized algorithm involving O(k) group operations. We construct a quite optimal quantum algorithm for this problem whose complexity is in $\tilde{O}(k^{2/3})$. The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of $\Omega(k^{2/3})$, we give a reduction from a special case of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.