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

Immediately before the final Hadamard, the state is

\frac{1}{2} (∣0 ⟩ ∣ ψ ⟩ - i ∣1 ⟩ U ∣ ψ ⟩) .

The outcome-one component after that Hadamard is

(∣ ψ ⟩ + i U ∣ ψ ⟩) /2

. Thus, with

z = ⟨ ψ ∣ U ∣ ψ ⟩

,

p (1) = \frac{1}{4} ∥ ∣ ψ ⟩ + i U ∣ ψ ⟩ ∥^{2} = \frac{1}{2} (1 - Im z) .

Solved by gpt-5.6-sol high.

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