Summary: Suppose we have \(n\) algorithms, quantum or classical, each computing some bit-value with bounded error probability. We describe a quantum algorithm that uses \(O(\sqrt{n})\) repetitions of the base algorithms and with high probability finds the index of a 1-bit among these \(n\) bits (if there is such an index). This shows that it is not necessary to first significantly reduce the error probability in the base algorithms to \(O(1/\text{poly}(n))\) (which would require \(O(\sqrt{n}\log n)\) repetitions in total). Our technique is a recursive interleaving of amplitude amplification and error-reduction, and may be of more general interest. Essentially, it shows that quantum amplitude amplification can be made to work also with a bounded-error verifier. As a corollary we obtain optimal quantum upper bounds of \(O(\sqrt{N})\) queries for all constant-depth AND-OR trees on \(N\) variables, improving upon earlier upper bounds of \(O(\sqrt{N}\text{poly}\log(N))\).
For the entire collection see [Zbl 1029.00041].