Pauli group

The Möbius–Kantor graph, the Cayley graph of the Pauli group [math]\displaystyle{ G_1 }[/math] with generators X, Y, and Z

In physics and mathematics, the Pauli group [math]\displaystyle{ G_1 }[/math] on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix [math]\displaystyle{ I }[/math] and all of the Pauli matrices

[math]\displaystyle{ 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} }[/math],

together with the products of these matrices with the factors [math]\displaystyle{ \pm 1 }[/math] and [math]\displaystyle{ \pm i }[/math]:

[math]\displaystyle{ 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 }[/math].

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

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

As an abstract group, [math]\displaystyle{ G_1\cong C_4 \circ D_4 }[/math] 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 [math]\displaystyle{ \sigma_1\sigma_2\sigma_3=iI }[/math] whereas there is no such relationship for the gamma group.

## References

* Nielsen, Michael A.; Chuang, Isaac L. (2000). Quantum Computation and Quantum Information. Cambridge; New York City: Cambridge University Press. ISBN 978-0-521-63235-5. OCLC 43641333.

## External links

1. ↑ Pauli group on GroupNames

2\. https://arxiv.org/abs/quant-ph/9807006

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