Test-State Approach to the Quantum Search Problem

Abstract: The search for "a quantum needle in a quantum haystack" is a metaphor for the problem of finding out which one of a permissible set of unitary mappings---the oracles---is implemented by a given black box. Grover's algorithm solves this problem with quadratic speed-up as compared with the analogous search for "a classical needle in a classical haystack." Since the outcome of Grover's algorithm is probabilistic---it gives the correct answer with high probability, not with certainty---the answer requires verification. For this purpose we introduce specific test states, one for each oracle. These test states can also be used to realize "a classical search for the quantum needle" which is deterministic---it always gives a definite answer after a finite number of steps---and faster by a factor of 3.41 than the purely classical search. Since the test-state search and Grover's algorithm look for the same quantum needle, the average number of oracle queries of the test-state search is the classical benchmark for Grover's algorithm.