nLab Pauli group

Definition

The Pauli group $\mathcal{P}_1$ is the finite group of order 16 which is isomorphic to the subgroup of the complex general linear group $GL(2,\mathbb{C})$ that consists of the multiples by $\pm 1$ and $\pm \mathrm{i}$ (where “$\mathrm{i}$” denotes the imaginary unit) of the four Pauli matrices $\sigma_{0,1,2,3}$:

(1)$$\mathcal{P}_1 \;\coloneqq\; \big\{ \pm \sigma_0, \, \pm \sigma_1, \, \pm \sigma_2, \, \pm \sigma_3, \,\; \pm i \sigma_0, \, \pm i \sigma_1, \, \pm i \sigma_2, \, \pm i \sigma_3, \big\} \;\subset\; GL(2,\mathbb{C}) \,,$$

where

$$\sigma_0 \;\coloneqq\; \left[ \array{ 1 & 0 \\ 0 & 1 } \right] ,\,\; \sigma_1 \;\coloneqq\; \left[ \array{ 0 & 1 \\ 1 & 0 } \right] ,\,\; \sigma_2 \;\coloneqq\; \left[ \array{ 1 & 0 \\ 0 & -1 } \right] ,\,\; \sigma_3 \;\coloneqq\; \left[ \array{ 0 & -\mathrm{i} \\ \mathrm{i} & 0 } \right] \,.$$

In quantum information theory one often considers the “higher” or “$n$-qbit” Pauli groups $\mathcal{P}_n$ whose elements are (multiples by $\pm 1$, $\pm \mathrm{i}$) of $n$-fold tensor products of the Pauli matrices.

The normalizer subgroup of the $n$-qbit Pauli group inside the unitary group $U(2^n)$ is known as the corresponding Clifford group.

Properties

Relation to the quaternion group

The quaternion group $Q$ is the (normal) subgroup of the Pauli group (1) omitting the $\pm 1$-phases on nonidentity matrices:

(2)$$Q \;\simeq\; \big\{ \pm \sigma_0, \, \pm \mathrm{i}\sigma_1, \, \pm \mathrm{i}\sigma_2, \, \pm \mathrm{i}\sigma_3, \big\} \;\subset\; GL(2,\mathbb{C}) \,,$$

In fact, the Pauli group is a semidirect product group of the quaternion group with the cyclic group of order 2:

$$\mathcal{P}_1 \;\simeq\; Q \rtimes \mathbb{Z}/2 \,.$$

Related concepts

• stabilizer code

• Clifford group

References

In the context of quantum computing:

• Michael A. Nielsen, Isaac L. Chuang, around (10.81) of: Quantum computation and quantum information, Cambridge University Press (2000, 2010) [doi:10.1017/CBO9780511976667, pdf, pdf]

See also:

• Wikipedia, Pauli group

Discussion in the context of quantum error correction codes:

• Simeon Ball, Aina Centelles, Felix Huber, p. 8 of: Quantum error-correcting codes and their geometries (arXiv:2007.05992)