Quantum phase estimation

Given controlled powers of a unitary $U$ and an eigenstate $U|\psi ⟩=e^{2\pi i\varphi }|\psi ⟩$, quantum phase estimation outputs an approximation to $\varphi$. If the input is a superposition of eigenstates, measurement selects an eigenphase with probability equal to the squared magnitude of its coefficient.

If $U|\psi ⟩=e^{2\pi ij/N}|\psi ⟩$, controlled powers of $U$ applied to a uniform $N$-state control register produceThese states are the Fourier basis. Applying the inverse quantum Fourier transform and measuring returns $j$ with probability one.