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

${\displaystyle X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}}$,

together with the products of these matrices with the factors ${\displaystyle \pm 1}$ and ${\displaystyle \pm i}$:

${\displaystyle G_{1}\ {\stackrel {\mathrm {def} }{=}}\ \{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv \langle X,Y,Z\rangle }$.

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

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

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

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

As an abstract group, ${\displaystyle G_{1}\cong C_{4}\circ D_{4}}$ 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 ${\displaystyle \sigma _{1}\sigma _{2}\sigma _{3}=iI}$ whereas there is no such relationship for the gamma group.

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

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

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