Table of Contents
Pauli gates
Pauli gates are the four single-qubit gates $I$, $X$, $Y$, $Z$ corresponding to the identity and the three Pauli matrices. They form the basis of single-qubit quantum operations and appear throughout quantum error correction, where $X$, $Y$, and $Z$ represent the three fundamental error types a qubit can suffer.
$$I = \begin{pmatrix}1 & 0\\0 & 1\end{pmatrix}\qquad X = \begin{pmatrix}0 & 1\\ 1 & 0\end{pmatrix}\qquad Y = \begin{pmatrix}0 & -i \\ i & 0\end{pmatrix}\qquad Z = \begin{pmatrix}1 & 0 \\ 0 & -1\end{pmatrix}$$
Algebraic properties
All four Pauli gates are both Hermitian ($P^\dagger = P$) and unitary ($P^\dagger P = I$), which means each one is its own inverse. Squaring any Pauli gate returns the identity:
$$I^2 = X^2 = Y^2 = Z^2 = I$$
The three non-identity Paulis anticommute with each other — multiplying in reverse order introduces a minus sign:
$$XY = -YX \qquad YZ = -ZY \qquad ZX = -XZ$$
Multiplying two distinct Paulis produces the third, up to a factor of $i$:
$$XY = iZ \qquad YZ = iX \qquad ZX = iY$$
Bloch sphere interpretation
On the Bloch sphere, each Pauli gate corresponds to a rotation of $\pi$ radians about the respective axis. The $X$ gate rotates about the $x$-axis (bit flip), $Z$ rotates about the $z$-axis (phase flip), and $Y$ rotates about the $y$-axis (both simultaneously, with a factor of $i$). The identity $I$ does nothing.