Pauli Group

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

Matrix group

The Pauli group consists of the 2 × 2 identity matrix $I$ and all of the Pauli matrices
:$X = \sigma_1 = \begin 0&1\\ 1&0 \end,\quad Y = \sigma_2 = \begin 0&-i\\ i&0 \end,\quad Z = \sigma_3 = \begin 1&0\\ 0&-1 \end$,
together with the products of these matrices with the factors $\pm 1$ and $\pm i$:
:$G \ \stackrel\ \ \equiv \langle X, Y, Z \rangle$.
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.

As an abstract group, $G \ \cong C_4 \circ D_4$ is the central product of a cyclic group of order 4 and the dihedral group of order 8.

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.

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.L. E. Dickson (1914) ''Linear Algebras'', pages 13,4 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 $n$ qubits, $G_n$, is the group generated by the operators described above applied to each of $n$ qubits in the tensor product Hilbert space $(\mathbb^2)^$. That is,

$G_n = \langle W_1 \otimes \cdots \otimes W_n : W_i \in \ \rangle.$

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.

References

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External links

Finite groups
Quantum information science

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