Pauli group

In physics and mathematics, the Pauli group is a 16-element matrix group

The Möbius–Kantor graph, the Cayley graph of the Pauli group with generators X, Y, and Z

Matrix group

The Pauli group consists 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 ${\displaystyle \pm i}$:

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

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

As an abstract group, ${\displaystyle G\ \cong C_{4}\circ D_{4}}$ 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.

Pauli algebra

The Pauli algebra is the algebra of 2 x 2 complex matrices M(2, C) with matrix addition and matrix multiplication. It has a long history beginning with the biquaternions introduced by W. R. Hamilton in his Lectures on Quaternions (1853). The representation with matrices was noted by L. E. Dickson in 1914.^[2] Publications by Pauli eventually led to the eponym now in use. Basis elements of the algebra generate the Pauli group.

Quantum computing

Quantum computing is based on qubits. 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,

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

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.

References

• 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.