Projector phase rotation
= Projector phase rotation
{title2=$e^{-i\theta P}=I+(e^{-i\theta}-1)P$}
For an <orthogonal projection>, $P^k=P$ for $k\geq1$. Summing the exponential series proves the displayed identity. At $\theta=\pi$, the operator is $I-2P$, which changes the sign on the range and fixes its orthogonal complement. In <Grover search algorithm>, the marked-state reflection and the uniform-state reflection compose to the search iterate up to an irrelevant <global phase>.