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(k2/3 ). The algorithm uses and highlights the power of the quantization method of Szegedy. For the lower bound of Ω(k2/3 ), 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.

• Type: Conference
• Year: 2005
• Where: ICALP
• Authors: Frédéric Magniez, Ashwin Nayak
• Keywords: Algorithm, ICALP 2005, Optimal Quantum Algorithm, Randomized Algorithm, Theoretical Computer Science