Hadamard test

The Hadamard test estimates the real part of $⟨\psi |U|\psi ⟩$ from one ancilla measurement.

If $U|\psi ⟩=e^{2\pi i\theta }|\psi ⟩$, the ordinary Hadamard test returns ancilla zero with probability

Adding a phase gate to the Hadamard test rotates the interference signal and measures the imaginary part of $⟨\psi |U|\psi ⟩$.