 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-4/10d/b/solution

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

Use

n

controlled-NOT gates in parallel. For each

j = 1, \dots, n

, the

j

th input qubit is the control and the

j

th output-register qubit is the target. Their combined action is

∣ x_{1} \dots x_{n} ⟩ ∣ y_{1} \dots y_{n} ⟩ ⟼ ∣ x_{1} \dots x_{n} ⟩ ∣ y_{1} \oplus x_{1}, \dots, y_{n} \oplus x_{n} ⟩,

which is exactly

∣ x ⟩ ∣ y ⟩ \mapsto ∣ x ⟩ ∣ y \oplus x ⟩ = U_{I} ∣ x ⟩ ∣ y ⟩ .

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