Amplitude amplification

If a state has good amplitude $\sin \theta$, the product of the reflection in the bad axis and the reflection in the state line rotates its good-bad plane by $2\theta$. After $k$ iterations the good probability isso $O(1/\sin \theta )$ iterations raise a small good amplitude close to one.

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.

For known good probability $p={\sin }^2\theta$, choose $j=\lceil \pi /(4\theta )-1/2\rceil$. A known reversible preparation of one state with good probability $p_{\ast }$ permits $j$ ordinary amplitude amplification iterations to reach the good subspace exactly. With ancillary marking access, prepare a flag with one-probability $q=p_{\ast }/p$ and define joint success as good data and flag one. Reflection about the prepared product state is known, while the joint phase test requires a controlled marking oracle or a computed Boolean output. This changes the success projector and enlarges the state space; it does not evade the obstruction to uniform probability lowering by a unitary. Query costs of preparation, its inverse and marking must all be counted.

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.

A geometric amplitude-amplification schedule tries iteration counts $1,2,4,8,\dots$ when the initial good amplitude is unknown. It reaches the unknown optimal scale with only a geometric-series overhead.