Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 324 / 2 / b / iv / Solution

An injective self-map of the finite set ${\mathbb{Z}}_N$ is a bijection. For sufficiently large $N$, exactlyoutputs satisfy the integer comparison, so there are exactly $m$ good inputs. Prepare the uniform superposition state $|\psi ⟩=N^{-1/2}{\sum }_x|x⟩$, whose good probability isUse the phase oracle from the previous part and the reflection $2|\psi ⟩⟨\psi |-I$. The amplitude amplification theorem requires $O(p^{-1/2})$ iterations; each uses two $U_f$ queries, while the state preparation and its reflection are independent of $f$. With the nearest-integer iteration count from that theorem, the failure probability is at most $p$, tending to zero. Therefore the success probability eventually exceeds $0.9$, withThis is a bound on quantum query complexity; implementation costs of the known gates are not counted as oracle queries.