SWAP exchanges the states of two qubits. If qubit 1 is in state $|a\rangle$ and qubit 2 is in state $|b\rangle$, after SWAP they exchange.

Matrix:

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

Action: $|ab\rangle \to |ba\rangle$ (swap basis state labels).

• Self-inverse: $\text{SWAP}^2 = I$
• Symmetric: swaps two qubits
• Reduces to identity on identical qubits (trivial for two copies of same state)

SWAP can be built from three CNOT gates:

$$\text{SWAP} = \text{CX}_{01} \text{CX}_{10} \text{CX}_{01}$$

This decomposition is commonly used since CNOT is more readily available.

• Qubit mapping: adjust layout to match hardware connectivity; move qubits closer for two-qubit gates
• Limited connectivity: enable operations between distant qubits on sparse graphs (at cost of extra gates)
• Circuit reordering: logically swap qubits to reduce circuit depth
• Quantum simulation: SWAP networks simulate particle dynamics on lattice geometries

• Expensive: requires 3 CNOT gates (vs 1 for controlled operations)
• Avoidance: redesign algorithms to avoid SWAP via gate reordering
• Necessary: chips like 1D chains require SWAPs to implement arbitrary algorithms

• Superconducting qubits: three CNOT pulses; gate time ~60–300 ns; some platforms have dedicated SWAP
• Trapped ions: composed from CX gates
• Cost note: dedicated SWAP implementations faster than three CNOTs