SWAP gate
| Identifier | Operator | Example statement |
|---|---|---|
| SWAP | \(SWAP\) | SWAP q[0], q[1] |
Description
The SWAP gate is a two-qubit gate. It swaps the state of the two qubits.
Properties
- Clifford gate;
- Involutory operation (its own inverse).
Representation
\[\begin{align}
SWAP &= \left(\begin{matrix}
1 & 0 & 0 & 0 \\
0 & 0 & 1 & 0 \\
0 & 1 & 0 & 0 \\
0 & 0 & 0 & 1
\end{matrix}\right)
\end{align}\]
Operation examples
Standard basis
\[\begin{align}
SWAP\,|00\rangle &= |00\rangle \\
\\
SWAP\,|01\rangle &= |10\rangle \\
\\
SWAP\,|10\rangle &= |01\rangle \\
\\
SWAP\,|11\rangle &= |11\rangle \\
\end{align}\]
Qubit state ordering convention and matrix representation
Note that qubits in a ket are ordered with qubit indices decreasing from left to right, i.e.,
\[|\psi\rangle = \sum c_i~|q_nq_{n-1}~...q_1q_0\rangle_i\]
Note that for matrices a reversed basis ordering convention is adopted, as is done in most textbooks. For instance, in the case of a two-qubit control gate, the matrix is represented such that \(q_0\) is the control qubit and \(q_1\) is the target qubit; the state on which the matrix is applied should then effectively be written as \(|q_0\rangle \otimes |q_1\rangle\).