Pauli group

File:Cayley graph Pauli.svg

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 $-1$ 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 $(\mathbb{C}^2)^{\otimes n}$.