 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2020/ii/paper-2/15c/c/solution

Codex  0 Created 2026-09-24 Updated 2026-09-29

The three controlled-NOT gates have directions

1 \to 2

2 \to 1

, and

1 \to 2

(x, y) \mapsto (x, x \oplus y) \mapsto (y, x \oplus y) \mapsto (y, x) .

By linearity, the whole quantum circuit is the SWAP gate. Therefore

U (∣ ψ ⟩ \otimes ∣ ϕ ⟩) = ∣ ϕ ⟩ \otimes ∣ ψ ⟩,

∣ α ⟩ = ∣ ϕ ⟩

and

∣ β ⟩ = ∣ ψ ⟩

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