Grover's search with faults on some marked elements

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 \cite{Gro96}. The running time of the algorithm, however, is very sensitive to errors in queries. It is known that if query may fail (report all marked elements as unmarked) the algorithm needs $\Omega(N)$ queries to find a marked element \cite{RS08}. \cite{AB+13} have proved the same result for the model where each marked element has its own probability to be reported as unmarked. 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 non-faulty marked items 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 limiting state $\rho_{lim}$.