Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 324 / 2 / a

Solution

Let

Π_{G}

∣ ψ ⟩ = sin θ ∣ ψ_{G} ⟩ + cos θ ∣ ψ_{B} ⟩, sin^{2} θ = ⟨ ψ ∣ Π_{G} ∣ ψ ⟩,

where the two displayed states are normalized projections into

G

and

G^{⊥}

R_{G} = I - 2 Π_{G}, R_{ψ} = 2∣ ψ ⟩ ⟨ ψ ∣ - I .

Q = R_{ψ} R_{G}

preserves the good-bad plane, and its

k

th iterate satisfies

Q^{k} ∣ ψ ⟩ = sin ((2 k + 1) θ) ∣ ψ_{G} ⟩ + cos ((2 k + 1) θ) ∣ ψ_{B} ⟩ .

Thus repeated reflections rotate amplitude toward the good subspace, reaching constant success probability after

O (1/ sin θ)

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