Exchange Gate

Definition¹

$2$Qubit $\ket{a, b} = \ket{a} \otimes \ket{b}$ The exchange gate $\text{ex}$ is defined as follows.

$$ \begin{align*} \text{ex} : (\mathbb{C}^{2})^{\otimes 2} &\to (\mathbb{C}^{2})^{\otimes 2} \\ \ket{a, b} &\mapsto \ket{b, a},\quad \forall a,b \in \left\{ 0, 1 \right\} \end{align*} $$

$$ \text{ex} (\ket{a} \otimes \ket{b}) = \ket{b} \otimes \ket{a} $$

Explanation

The exchange gate swaps the states of two qubits. The specific input and output are as follows.

$$ \text{ex} (\ket{00}) = \ket{00} \\[0.5em] \text{ex} (\ket{01}) = \ket{10} \\[0.5em] \text{ex} (\ket{10}) = \ket{01} \\[0.5em] \text{ex} (\ket{11}) = \ket{11} $$

Matrix representation is as follows.

$$ \text{ex} = \begin{bmatrix} 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 0 & 1 \end{bmatrix} $$