SWAP gate

SWAP exchanges the states of two qubits: $\mathrm{SWAP}|a,b\rangle=|b,a\rangle$ for every computational basis input, exactly the check this record performs across all four inputs. As a matrix, $\mathrm{SWAP}=\begin{pmatrix}1&0&0&0\\0&0&1&0\\0&1&0&0\\0&0&0&1\end{pmatrix}$ in the $\{|00\rangle,|01\rangle,|10\rangle,|11\rangle\}$ basis, and $\mathrm{SWAP}^2=I$. Algebraically SWAP decomposes into three CNOTs, $\mathrm{SWAP}=\mathrm{CX}_{12}\,\mathrm{CX}_{21}\,\mathrm{CX}_{12}$, which is what a compiler emits when SWAP is not a native two-qubit interaction. SWAP does not create or destroy entanglement between the qubits it moves; it relocates existing correlations to different wires. In practice SWAP is a routing/connectivity primitive: transpilers insert it to move a logical qubit's state across a hardware coupling graph so a two-qubit gate can be applied between qubits that are not physically adjacent, at the cost of extra circuit depth.