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.

Phase-matched amplitude amplification replaces the two sign-flip reflections by selective phase rotations. Choosing their phases from the known initial overlap can rotate the state exactly onto the target even when an integer number of ordinary Grover rotations would overshoot it.