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On the Quantum Black-Box Complexity of Majority

Thomas Hayes·Samuel Kutin·Dieter van Melkebeek
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Abstract

We describe a quantum black-box network computing the majority of N bits with zero-sided error eps using only 2N/3 + O(sqrt{N (log log N + log 1/eps)}) queries: the algorithm returns the correct answer with probability at least 1 - eps, and "I don't know" otherwise. Our algorithm is given as a randomized "XOR decision tree" for which the number of queries on any input is strongly concentrated around a value of at most 2N/3. We provide a nearly matching lower bound of 2N/3 - O(sqrt(N)) on the expected number of queries on a worst-case input in the randomized XOR decision tree model with zero-sided error o(1). Any classical randomized decision tree computing the majority on N bits with zero-sided error 1/2 has cost N.

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Thomas Hayes, Samuel Kutin, Dieter van Melkebeek. 2002-10-30. On the Quantum Black-Box Complexity of Majority. https://arxiv.org/abs/quant-ph/0109101

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