Pauli group

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

$X={\sigma }_1=\left(\begin{array}{cc}0& 1\\ 1& 0\end{array}\right),Y={\sigma }_2=\left(\begin{array}{cc}0& -i\\ i& 0\end{array}\right),Z={\sigma }_3=\left(\begin{array}{cc}1& 0\\ 0& -1\end{array}\right)$,

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

$G_1\stackrel{\mathrm{d}\mathrm{e}\mathrm{f}}{=}\{\pm I,\pm iI,\pm X,\pm iX,\pm Y,\pm iY,\pm Z,\pm iZ\}\equiv ⟨X,Y,Z⟩$.

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 $({ℂ}^2{)}^{\otimes n}$. That is,

$$G_n=⟨W_1\otimes \cdots \otimes W_n:W_i\in \{X,Y,Z\}⟩.$$

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.

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.

References

• Template:Cite book

External links

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