  In this work we address two questions concerning Grover's algorithm. In the
first we give an answer to the question how to employ Grover's algorithm for
actual search over database. We introduce a quantum model of an unordered phone
book (quantum database) with programmable queries to search in the phone book
either for a number, or for a name. In the second part we investigate how
successful the algorithm can be if the number of elements of the database is
not known precisely. This question reduces to analysis of the
distinguishability of states occurring during Grover's algorithm. We found that
using unambiguous discrimination scheme even a seemingly good guess, that is
close to the optimal one can result in a rather small success rate.
