Optimizing the number of gates in quantum search

In its usual form, Grover’s quantum search algorithm uses O( √ N) queries and O( √ N logN) other elementary gates to find a solution in an N -bit database. Grover in 2002 showed how to reduce the number of other gates to O( √ N log logN) for the special case where the database has a unique solution, without significantly increasing the number of queries. We show how to reduce this further to O( √ N log N) gates for every constant r, and sufficiently large N . This means that, on average, the circuits between two queries barely touch more than a constant number of the logN qubits on which the algorithm acts. For a very large N that is a power of 2, we can choose r such that the algorithm uses essentially the minimal number π 4 √ N of queries, and only O( √ N log(logN)) other gates.