|q⟩ Bad Qubits

beginner · Programming · Controlled Gates: CNOT, CZ, SWAP

The SWAP Gate

The SWAP gate exchanges the complete quantum states of two qubits. Where CNOT conditionally flips one qubit, SWAP unconditionally trades the contents of both — whatever amplitude was on qubit aa moves to qubit bb, and vice versa.

What SWAP does

On basis states, SWAP simply permutes the two labels:

SWAP ∣00⟩=∣00⟩,SWAP ∣01⟩=∣10⟩,SWAP ∣10⟩=∣01⟩,SWAP ∣11⟩=∣11⟩.\text{SWAP}\,|00\rangle = |00\rangle, \quad \text{SWAP}\,|01\rangle = |10\rangle, \quad \text{SWAP}\,|10\rangle = |01\rangle, \quad \text{SWAP}\,|11\rangle = |11\rangle.

In the computational basis {∣00⟩,∣01⟩,∣10⟩,∣11⟩}\{|00\rangle, |01\rangle, |10\rangle, |11\rangle\} (qubit 0 most significant) this corresponds to the unitary matrix

SWAP=(1000001001000001).\text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}.

The off-diagonal 11s in the ∣01⟩|01\rangle/∣10⟩|10\rangle block are what effect the exchange.

SWAP on superpositions

Because SWAP is linear, it works just as cleanly on superpositions. If qubit 0 carries amplitude α∣0⟩+β∣1⟩\alpha|0\rangle + \beta|1\rangle and qubit 1 carries γ∣0⟩+δ∣1⟩\gamma|0\rangle + \delta|1\rangle, then after SWAP qubit 0 carries γ∣0⟩+δ∣1⟩\gamma|0\rangle + \delta|1\rangle and qubit 1 carries α∣0⟩+β∣1⟩\alpha|0\rangle + \beta|1\rangle — a full exchange of the amplitude packets, including any relative phases.

Try it

The circuit below puts qubit 0 into ∣1⟩|1\rangle, leaving the two-qubit register in ∣10⟩|10\rangle. Your task is to apply SWAP so the register ends in ∣01⟩|01\rangle — qubit 0 returns to ∣0⟩|0\rangle and qubit 1 receives the ∣1⟩|1\rangle.

Run your code to see the quantum state.

After a correct solution the statevector should show amplitude 11 on the ∣01⟩|01\rangle basis state and zero everywhere else.

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