Hadamard test

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

For the Hadamard test with a density operator $\rho$ in the auxiliary register, the output control qubit has Pauli Z gate expectation $\mathrm{Re}m$ and Pauli Y gate expectation $-\mathrm{Im}m$, where $m=\mathrm{Tr}(\rho U)$. The minus sign comes from the final Hadamard gate, since $HYH=-Y$. Measuring in $(|0⟩\pm i|1⟩)/\sqrt{2}$ therefore estimates the imaginary part using the minus-outcome frequency minus the plus-outcome frequency.

Prepare the auxiliary register in the maximally mixed state $I/N$ and estimate both mixed-state Hadamard-test quadratures. Multiplying the normalized expectation by $N$ estimates the trace of $U$. A uniformly randomized preparation of any orthonormal basis produces $I/N$; the eigenvectors of $U$ need not be known.

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 $⟨\psi |U|\psi ⟩$.