 Pauli Group
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Pauli Group The Möbius-Kantor graph, the Cayley graph of the Pauli group $G_{1}$ with generators X, Y, and Z

In physics and mathematics, the Pauli group $G_{1}$ on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix $I$ and all of the Pauli matrices

$X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ ,

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

$G_{1}\ {\stackrel {\mathrm {def} }{=}}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv \langle X,Y,Z\rangle$ .

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

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

As an abstract group, $G_{1}\cong C_{4}\circ D_{4}$ is the central product of a cyclic group of order 4 and the dihedral group of order 8.