Theorem: Hadamard-test P(0) = 1/2 + (1/2)*cos(phi); P(0)|_{phi=0} = 1; P(0)|_{phi=pi} = 0

Theorem (Hadamard-test cosine readout): the probability of measuring |0> on the ancilla after H-controlled-U-H with U|psi> = e^{i*phi}|psi> is P(0|phi) = 1/2 + (1/2)*cos(phi). Boundary canonicals: phi=0 (U acts as identity on eigenvector)…