= Known-state success dilution for exact amplitude amplification
{title2=$p_*=\sin^2[\pi/(4j+2)]\le p$}
= Success dilution
{synonym}
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 $p_*$ permits $j$ ordinary <amplitude amplification> iterations to reach the good subspace exactly. With ancillary marking access, prepare a flag with one-probability $q=p_*/p$ 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.
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