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Grover’s algorithm is a quantum query algorithm solving the unstructured search problem of size $N$ using $O(\sqrt{N})$ queries. It provides a significant speed-up over any classical algorithm [3].

The running time of the algorithm, however, is very sensitive to errors in queries. Multiple authors have analysed the algorithm using different models of query errors and showed the loss of quantum speed-up [2, 6].

We study the behavior of Grover’s algorithm in the model where the search space contains both faulty and non-faulty marked elements. We show that in this setting it is indeed possible to find one of marked elements in $O(\sqrt{N})$ queries.

We also analyze the limiting behavior of the algorithm for a large number of steps and show the existence and the structure of a limiting state ${\rho }_{lim}$.