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 (O) over tilde (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.