# Pauli operators(gates) and Pauli group These are the operators $I$,[[X-gate (NOT gate)|X]],[[Y-gate|Y]],[[Z-gate (Relative phase gate, Pauli gate)|Z]]. They generate the Pauli-group with anti-commute group operation. Each basic operator can be represented by $Z^aX^b$ for $a,b\in\{0,1\}$ up to [[Global phase]]: $ \begin{align} I &= Z^0X^0 \\ X &= Z^0X^1 \\ Z &= Z^1X^0 \\ Y &= -iZ^1X^1\equiv Z^1X^1 \end{align} $ This means that Pauli is generated by $Z$ and $X$. $Z$ and $X$ are anti-commutative i.e., $ZX=-XZ$ because: $(ZX)(\alpha|0\rangle+\beta|1\rangle)=-\alpha|1\rangle+\beta|0\rangle =-\big(\alpha|1\rangle-\beta|0\rangle\big) =(-XZ)(\alpha|0\rangle+\beta|1\rangle)$ This means that every operator in the Pauli group can be represented as $Z^aX^b$ : first represent every $I$ and $Y$ as powers of $Z$ and $X$, then use the anti-commutative property to collect $Z$ on one side and $X$ on the other. $n$-qubit Pauli is a tensor product of $n$ Pauli gates. for example: $X\otimes Z$. If $U$ in Pauli group it can be represented as a matrix $(a,b)\in\mathbb{F}_2^2$ i.e., $U=Z^aX^b$. ## Created 2026-02-22 12:43