An Upper Bound on the Rate of Information Transfer by Grover’s Oracle

Abstract

Grover discovered a quantum algorithm for identifying a target element in an unstructured search universe of N items in approximately $\pi/4 \sqrt{N}$ queries to a quantum oracle. For classical search using a classical oracle, the search complexity is of order N/2 queries since on average half of the items must be searched. In work preceding Grover’s, Bennett et al. had shown that no quantum algorithm can solve the search problem in fewer than $O(\sqrt{N})$ queries. Thus, Grover’s algorithm has optimal order of complexity. Here, we present an information-theoretic analysis of Grover’s algorithm and show that the square-root speed-up by Grover’s algorithm is the best possible by any algorithm using the same quantum oracle.