Pauli Gates

Definition¹

A qubit gate $X, Y, Z$, defined as follows, is called a Pauli gate.

$$ X, Y, Z : \mathbb{C}^{2} \to \mathbb{C}^{2} $$

$$ \begin{array}{l} X \ket{0} = \ket{1} \\ X \ket{1} = \ket{0} \end{array} \qquad \begin{array}{l} Y \ket{0} = -\ket{1} \\ Y \ket{1} = \ket{0} \end{array} \qquad \begin{array}{l} Z \ket{0} = \ket{0} \\ Z \ket{1} = -\ket{1} \end{array} $$

The matrix representations are as follows.

$$ X = \begin{bmatrix} 0 & 1 \\ 1 & 0 \end{bmatrix} \qquad Y = \begin{bmatrix} 0 & 1 \\ -1 & 0 \end{bmatrix} \qquad Z = \begin{bmatrix} 1 & 0 \\ 0 & -1 \end{bmatrix} $$

Explanation

Each matrix representation is the same as the Pauli matrices.

The Pauli $X$ gate swaps $\ket{0}$ with $\ket{1}$, and $\ket{1}$ with $\ket{0}$. In this aspect, it can be seen as a quantum version of the $\text{NOT}$ gate. Also, for a qubit $\alpha_{0}\ket{0} + \alpha_{1}\ket{1}$ in a superposition state, it changes the probabilities of being measured as $\ket{0}$ and $\ket{1}$ with each other.