Quantum complexity of testing group commutativity

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 Õ(k2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Ω(k2/3), we introduce a new technique of reduction for quantum query complexity. Along the way, we prove the optimality of the algorithm of Pak for the randomized model. © Springer-Verlag Berlin Heidelberg 2005.