, Volume 48, Issue 3, pp 221–232

Quantum Complexity of Testing Group Commutativity



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.


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Copyright information

© Springer 2007

Authors and Affiliations

  1. 1.CNRS, LRI - batiment 490, Universite Paris-Sud91405 Orsay cedexFrance
  2. 2.Department of Combinatorics and Optimization and Institute for Quantum Computing, University of Waterloo, 200 University Ave. W.Waterloo, Ontario N2L 3G1Canada

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