On the Quantum Black-Box Complexity of Majority
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.
Keywords
Cite
@article{arxiv.quant-ph/0109101,
title = {On the Quantum Black-Box Complexity of Majority},
author = {Thomas Hayes and Samuel Kutin and Dieter van Melkebeek},
journal= {arXiv preprint arXiv:quant-ph/0109101},
year = {2007}
}
Comments
22 pages, to appear in Algorithmica, v3: tail laws in appendix proved in a more elegant way than in the journal version