Pauli group



In physics and mathematics, the Pauli group $$G_1$$ on 1 qubit is the 16-element matrix group consisting of the 2 &times; 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.

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 $$\sigma_1\sigma_2\sigma_3=iI$$ whereas there is no such relationship for the gamma group.