 Exact amplitude amplification

If the initial good amplitude $p=\sin \theta$ is known, an ancillary qubit can reduce it to $\sin {\theta }^{\prime }=pc$ with ${\theta }^{\prime }=\pi /(4k+2)$ and $0<c\le 1$. After $k$ ordinary amplitude-amplification iterations the good amplitude is $\sin ((2k+1){\theta }^{\prime })=1$, so the target state is prepared exactly.