  We report the implementation of Grover's quantum search algorithm in the
scalable system of trapped atomic ion quantum bits. Any one of four possible
states of a two-qubit memory is marked, and following a single query of the
search space, the marked element is successfully recovered with an average
probability of 60(2)%. This exceeds the performance of any possible classical
search algorithm, which can only succeed with a maximum average probability of
50%.
