Pauli group

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 $(\mathbb {C} ^{2})^{\otimes n}$ . That is,

$G_{n}=\langle W_{1}\otimes \cdots \otimes W_{n}:W_{i}\in \{X,Y,Z\}\rangle .$

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

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.^[1]

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.