nLab Pauli group

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Definition

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

(1)𝒫 1{±σ 0,±σ 1,±σ 2,±σ 3,±iσ 0,±iσ 1,±iσ 2,±iσ 3,}GL(2,), \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

σ 0[1 0 0 1],σ 1[0 1 1 0],σ 2[1 0 0 1],σ 3[0 i i 0]. \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 “nn-qbit” Pauli groups 𝒫 n\mathcal{P}_n whose elements are (multiples by ±1\pm 1, ±i\pm \mathrm{i}) of nn-fold tensor products of the Pauli matrices.

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

Properties

Relation to the quaternion group

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

(2)Q{±σ 0,±iσ 1,±iσ 2,±iσ 3,}GL(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:

𝒫 1Q/2. \mathcal{P}_1 \;\simeq\; Q \rtimes \mathbb{Z}/2 \,.

References

In the comtext of quantum computing:

See also:

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)

Last revised on April 8, 2025 at 12:02:42. See the history of this page for a list of all contributions to it.