 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2025/ii/paper-1/10c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-25

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_{X I}

, then

⟨ ψ ∣ Z \otimes Z ∣ ψ ⟩ = 1 - 2 p_{ZZ}, ⟨ ψ ∣ X \otimes I ∣ ψ ⟩ = 1 - 2 p_{X I} .

The identity in part (b) implements each controlled tensor product by gates on its two target wires with the common control. By linearity,

⟨ ψ ∣ W ∣ ψ ⟩ = 2 - 2 (p_{ZZ} + p_{X I}) .

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