 Solution

ID: past-exam-of-the-laser-science-and-quantum-information-processing-course-of-the-university-of-oxford/2017/4/a/solution

Use the time-ordered network

CNOT_{1 \to 2}

CNOT_{2 \to 1}

CNOT_{1 \to 2}

(a, b)

its successive values are

(a, b) ⟼ (a, a \oplus b) ⟼ (b, a \oplus b) ⟼ (b, a) .

Thus the network exchanges the two computational-basis states. Since the gates are linear, this basis proof also establishes SWAP gate on arbitrary quantum states.

∣ Φ^{\pm} ⟩ = (∣00 ⟩ \pm ∣11 ⟩) / 2

and

∣ Ψ^{\pm} ⟩ = (∣01 ⟩ \pm ∣10 ⟩) / 2

obey

S ∣ Φ^{+} ⟩ = ∣ Φ^{+} ⟩, S ∣ Φ^{-} ⟩ = ∣ Φ^{-} ⟩, S ∣ Ψ^{+} ⟩ = ∣ Ψ^{+} ⟩, S ∣ Ψ^{-} ⟩ = - ∣ Ψ^{-} ⟩ .

The antisymmetric spin singlet state alone has eigenvalue

- 1

. Its standalone sign is a global phase; a Fredkin gate can make that sign observable relative to the control branch, as in part d.

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