Past exam of the mathematics course of the University of Cambridge / 2025 / ii / Paper 1 / 10C / c / Solution

Both $Z\otimes Z$ and $X\otimes I$ are unitary and Hermitian, so their expectations are real. Run the real-part Hadamard test twice, once with controlled $Z\otimes Z$ and once with controlled $X\otimes I$. If the respective ancilla outcome-one probabilities are $p_{ZZ}$ and $p_{XI}$, thenThe identity in part (b) implements each controlled tensor product by gates on its two target wires with the common control. By linearity,