Hadamard-test / phase-kickback: P(0) = 1/2 + (1/2)*cos(phi) from H-controlled-U-H on eigenvector

Hadamard-test / phase-kickback framework. Setup: ancilla qubit |0> and an eigenvector register in state |psi> with U|psi> = e^{i*phi}|psi>. Circuit: (1) Hadamard on ancilla: (1/sqrt(2))(|0> + |1>) |psi>; (2) controlled-U conditioned on…