  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 $\rho_{lim}$.
