Grover's algorithm with errors

Abstract

Grover's algorithm is a quantum search algorithm solving the unstructured search problem of size n in O(√n)queries, while any classical algorithm needs O(n) queries [3]. However, if query has some small probability of failing (reporting that none of the elements are marked), then quantum speed-up disappears: no quantum algorithm can be faster than a classical exhaustive search by more than a constant factor [8]. We study the behaviour of Grover's algorithm in the model there query may report some marked elements as unmarked (each marked element has its own error probability, independent of other marked elements). We analyse the limiting behaviour of Grover's algorithm for a large number of steps and prove the existence of limiting state ρ lim. Interestingly, the limiting state is independent of error probabilities of individual marked elements. If we measure ρ lim, the probability of getting one of the marked states i1, ... , ik is k+1/k. We show that convergence time is O(n). © 2013 Springer-Verlag.