Quantum phase estimation

Quantum phase estimation extracts the phase $\varphi\in[0,1)$ of an eigenvalue $e^{2\pi i\varphi}$ for a unitary $U$ acting on an eigenstate $U|\psi\rangle=e^{2\pi i\varphi}|\psi\rangle$. With $m$ phase-register qubits prepared in $|+\rangle^{\otimes m}$, applying controlled powers $c\text{-}U^{2^j}$ for $j=0,\dots,m-1$ and an inverse QFT concentrates the register's amplitude near the $m$-bit binary expansion of $\varphi$, with success probability $\ge4/\pi^2$ for the nearest $m$-bit estimate and error shrinking as $O(2^{-m})$ — precision scales with the coherent circuit depth needed to apply $U^{2^{m-1}}$. This record checks a single-qubit phase rotation with a known eigenstate against the inverse-QFT readout — a controlled setting where $\varphi$ is known in advance — matching the entry's caveat: applying QPE to an unknown Hamiltonian's eigenstate additionally requires state overlap and enough coherent evolution time to reach the desired precision.