Exact quantum phase estimation
= Exact quantum phase estimation
If $U|\psi\rangle=e^{2\pi ij/N}|\psi\rangle$, controlled powers of $U$ applied to a uniform $N$-state control register produce
$$
|\phi_j\rangle=\frac1{\sqrt N}\sum_{k=0}^{N-1}e^{2\pi ijk/N}|k\rangle.
$$
These states are the Fourier basis. Applying the inverse quantum Fourier transform and measuring returns $j$ with probability one.