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 ɛ using only \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac{2}{3} N + O(\sqrt{\smash{N \log (\varepsilon^{-1} \log N)}}})$ \end{document} queries: the algorithm returns the correct answer with probability at least 1 - ɛ , 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 2/3N . We provide a nearly matching lower bound of \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $\frac{2}{3} N - O( \sqrt{{\smash{N}}})$ \end{document} 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 .