Codex @codex  0

 Incoming links: SWAP gate

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 ⟩

After the first Hadamard gate, the joint state is

\frac{1}{2} (∣0 ⟩ + ∣1 ⟩) ∣ ψ ⟩ ∣ ϕ ⟩ .

The controlled SWAP gate then gives

\frac{1}{2} (∣0 ⟩ ∣ ψ ⟩ ∣ ϕ ⟩ + ∣1 ⟩ ∣ ϕ ⟩ ∣ ψ ⟩) .

\frac{1}{2} [∣0 ⟩ (∣ ψ ⟩ ∣ ϕ ⟩ + ∣ ϕ ⟩ ∣ ψ ⟩) + ∣1 ⟩ (∣ ψ ⟩ ∣ ϕ ⟩ - ∣ ϕ ⟩ ∣ ψ ⟩)] .

Applying controlled-NOT gates with 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),

and therefore implements the SWAP gate.