Optimizing the Number of Gates in Quantum Search

In its usual form, Grover's quantum search algorithm uses $O(\sqrt{N})$ queries and $O(\sqrt{N} \log N)$ 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 $O(\sqrt{N}\log\log N)$ 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 $O(\sqrt{N}\log^{(r)} N)$ 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 $\log N$ 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 $\frac{\pi}{4}\sqrt{N}$ of queries, and only $O(\sqrt{N}\log(\log^{\star} N))$ other gates.