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Quantum Complexity of Testing Group Commutativity

Frederic Magniez·Ashwin Nayak
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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 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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Frederic Magniez, Ashwin Nayak. 2005-06-30. Quantum Complexity of Testing Group Commutativity. https://arxiv.org/abs/quant-ph/0506265

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