Past exam of the mathematics course of the University of Cambridge / 2025 / iii / Paper 324 / 2 / ii / Solution

Normalization gives $p^2+q^2=1$. Write $p=\sin \theta$, $q=\cos \theta$ with $0<\theta <\pi /2$, and set $|u⟩=U|b⟩$. Define the two Householder reflectionsThe amplitude amplification iterate $G=R_u{R}_{\xi }$ preserves $\mathrm{span}\{|\xi ⟩,|\varphi ⟩\}$ and rotates that plane through $2\theta$. ConsequentlyIn particular, $O(1/p)$ iterations raise the success probability to a constant close to one.