The SWAP Gate

The SWAP Gate

The SWAP gate exchanges the states of two qubits:

$\text{SWAP}|01\rangle = |10\rangle \qquad \text{SWAP}|10\rangle = |01\rangle$

It leaves $|00\rangle$ and $|11\rangle$ unchanged.

Matrix Representation

$\text{SWAP} = \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

State Vector Indexing

A 2-qubit system has 4 basis states indexed as $q_0 \cdot 2 + q_1$:

SWAP exchanges the amplitudes at indices 1 and 2 — swapping $|01\rangle \leftrightarrow |10\rangle$.

Decomposition into CNOTs

The SWAP gate can be decomposed into three CNOT gates:

$\text{SWAP} = \text{CNOT}_{01} \cdot \text{CNOT}_{10} \cdot \text{CNOT}_{01}$

This decomposition is useful in hardware where direct qubit-qubit connectivity is limited.

Self-Inverse

Like CNOT and Toffoli, SWAP is its own inverse:

$\text{SWAP}^2 = I$

Applications

• Qubit routing: moving quantum information between non-adjacent qubits in hardware
• Quantum sorting networks: sorting qubit registers
• iSWAP: a variant $\text{iSWAP}|01\rangle = i|10\rangle$ used in superconducting processors

Implementation

def swap_gate(state):
    result = list(state)
    result[1], result[2] = state[2], state[1]
    return result

Your Task

Implement swap_gate(state) for a 4-element 2-qubit state vector that swaps the amplitudes of $|01\rangle$ (index 1) and $|10\rangle$ (index 2).