Simplified quantum algorithm for the oracle identification problem
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Abstract
In the oracle identification problem we have oracle access to bits of an unknown string x of length n, with the promise that it belongs to a known set C⊆{0,1}n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$C\subseteq \{0,1\}^{n}$$\end{document}. The goal is to identify x using as few queries to the oracle as possible. We develop a quantum query algorithm for this problem with query complexity OnlogMlog(n/logM)+1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$O\left (\sqrt {\frac {n\log M }{\log (n/\log M)+1}}\right )$$\end{document}, where M is the size of C. This bound is already derived by Kothari in 2014, for which we provide a more elegant simpler proof.