Pauli group explained

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 matricesX = \sigma _{1} = \begin{pmatrix} 0 & 1 \\ 1 & 0 \end{pmatrix}, Y = \sigma _{2} = \begin{pmatrix} 0 & - i\\ i& 0 \end{pmatrix}, Z = \sigma _{3} = \begin{pmatrix} 1 & 0 \\ 0 & -1 \end{pmatrix} ,together with the products of these matrices with the factors \pm 1 and \pm i :G _{1} \stackrel{ def }\ \ \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 ( C ^{2)} ^{⊗} . That is,

$$G\_n\; =\; \backslash langle\; W\_1\; \backslash otimes\; \backslash cdots\; \backslash otimes\; W\_n\; :\; W\_i\; \backslash in\; \backslash \; \backslash rangle.$$

The order of

G _{n} is 4 ⋅ 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.^[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

\sigma _{1\sigma} _{2\sigma} _{3=iI} whereas there is no such relationship for the gamma group.

References

• Book: Quantum Computation and Quantum Information. Nielsen. Michael A.. Michael Nielsen . Chuang . Isaac L. . Isaac Chuang. 2000. Cambridge University Press. Cambridge

  New York

  . 978-0-521-63235-5. 43641333.

External links

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

Notes and References

1. Pauli group on GroupNames