On the Quantum Black-Box Complexity of Majority

Details On the Quantum Black-Box Complexity of Majority On the Quantum Black-Box Complexity of Majority

2001-09-20

arxiv.org/abs/quant-ph/0109101v3

Physics

Quantum Physics

22 pages, to appear in Algorithmica, v3: tail laws in appendix proved in a more elegant way than in the journal version

Scientific paper

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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