Pauli group

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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}.

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