Pauli group

${\displaystyle X={\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),Y={\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),Z={\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)}$,

together with the products of these matrices with the factors ${\displaystyle \pm 1}$ and ${\displaystyle \pm i}$:

${\displaystyle G_1\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv ⟨X,Y,Z⟩}$.

The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.

The Pauli group on ${\displaystyle n}$ qubits, ${\displaystyle G_n}$, is the group generated by the operators described above applied to each of ${\displaystyle n}$ qubits in the tensor product Hilbert space ${\displaystyle ({\mathbb{C}}^2{)}^{\otimes n}}$. That is,

$${\displaystyle G_n=⟨W_1\otimes \cdots \otimes W_n:W_i\in \{X,Y,Z\}⟩.}$$

The order of ${\displaystyle G_n}$ is ${\displaystyle 4\cdot 4^n}$ since a scalar ${\displaystyle \pm 1}$ or ${\displaystyle \pm i}$ factor in any tensor position can be moved to any other position.

As an abstract group, ${\displaystyle G_1\cong C_4\circ D_4}$ is the central product of a cyclic group of order 4 and the dihedral group of order 8.

The Pauli group is a representation of the gamma group in three-dimensional Euclidean space. It is not isomorphic to the gamma group; it is less free, in that its chiral element is ${\displaystyle {\sigma }_1{\sigma }_2{\sigma }_3=iI}$ whereas there is no such relationship for the gamma group.