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.
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Magniez, F., Nayak, A. Quantum Complexity of Testing Group Commutativity. Algorithmica 48, 221–232 (2007). https://doi.org/10.1007/s00453-007-0057-8
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DOI: https://doi.org/10.1007/s00453-007-0057-8