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(k 2/3). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Ω(k 2/3), 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.
For their research support, F.M. thanks the EU 5th framework program RESQ and the French Research Ministry, and A.N. thanks Canada’s NSERC and CIAR.
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Magniez, F., Nayak, A. (2005). Quantum Complexity of Testing Group Commutativity. In: Caires, L., Italiano, G.F., Monteiro, L., Palamidessi, C., Yung, M. (eds) Automata, Languages and Programming. ICALP 2005. Lecture Notes in Computer Science, vol 3580. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11523468_106
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DOI: https://doi.org/10.1007/11523468_106
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