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ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/ii/paper-2/15d/b/i/solution

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Past exam of the mathematics course of the University of Cambridge / 2022 / ii / Paper 2 / 15D / b / i / Solution by

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

For two nonorthogonal input states

∣ α ⟩, ∣ β ⟩

, preservation of inner products by

G

gives

⟨ α ∣ β ⟩ = ⟨ α ∣ β ⟩ ⟨ P_{U} (α) ∣ P_{U} (β)⟩ .

(1)

Hence the two output-program states agree up to phase. Any two qubit states can be joined through a state nonorthogonal to both, so after absorbing phases

∣ P_{U} ⟩

is independent of the input.

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