 Solution

If some integer $k$ obeys $(2k+1)\theta =\pi /2$, ordinary amplitude amplification already gives $|g⟩$ exactly. For a general known $\theta$, use exact amplitude amplification. Choose $k$ so thatand put $c=\sin {\theta }^{\prime }/\sin \theta \le 1$. Append an ancilla and coherently arrange that its designated good value has amplitude $c$ conditional on the original register being good. With the enlarged good subspace defined bythe starting state's total good amplitude is $c\sin \theta =\sin {\theta }^{\prime }$.

Apply the amplitude amplification theorem $k$ times to this enlarged problem. Its final good amplitude isThe Boolean quantum oracle implements the reflection $I_G$ by phase kickback, and the known state-preparation circuit implements the reflection about the starting state by prepare--reflect--unprepare. A final computational-basis measurement therefore yields an $x$ with $f(x)=1$ with certainty.