Quantum Complexity of Testing Group Commutativity [article]

Frederic Magniez, Ashwin Nayak (U. Waterloo and Perimeter Inst.)
2007 arXiv   pre-print
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 (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
more » ... ase of Element Distinctness to our problem. Along the way, we prove the optimality of the algorithm of Pak for the randomized model.
arXiv:quant-ph/0506265v4 fatcat:sfknwj2apbdrfpamh2u5iis4mm