 Solution

Let $t=e^{iz}$ and $|u⟩=U|b⟩=p|\xi ⟩+q|\varphi ⟩$. The first factor in $G$ isSince $R(\xi ,z)|u⟩=pt|\xi ⟩+q|\varphi ⟩$, the coefficient of $|\varphi ⟩$ in $G|u⟩$ isIt vanishes whenor, using $p^2+q^2=1$,A unit-modulus solution $t=e^{iz}$ exists exactly when the right-hand side lies in $[-2,2]$. The upper bound is automatic, while the lower bound isThus exact preparation by one application of $G$ is possible precisely whenOne may choose $z$ so that $\cos z=(p^2-q^2)/(2p^2)$. Then $G|u⟩$ has no $|\varphi ⟩$ component and, by unitarity, equals $|\xi ⟩$ up to phase. This is a phase-matched amplitude amplification step.