In physics, quantum information and group theory, the Pauli group is a group formed by tensor products of Pauli matrices, including the identity. The single-qubit Pauli group is a 16-element matrix group, consisting of the 4 Pauli matrices each with 4 possible phase factors. The n-qubit Pauli group is a 4 n + 1 {\displaystyle 4^{n+1}} -element group consisting of tensor products of single-qubit Paulis. In quantum information theory, Pauli groups are important because they are the basis for stabilizer formalism, a widely-used framework for constructing and describing quantum error correction codes using sets of commuting Pauli operators. Stabilizer codes are formed from commuting subgroups of the Pauli group.
Single-qubit Pauli group The Pauli group consists of the 2 × 2 identity matrix I {\displaystyle I} and all of the Pauli matrices
X = σ 1 = ( 0 1 1 0 ) , Y = σ 2 = ( 0 − i i 0 ) , Z = σ 3 = ( 1 0 0 − 1 ) {\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 ± 1 {\displaystyle \pm 1} and ± i {\displaystyle \pm i} :
G = d e f { ± I , ± i I , ± X , ± i X , ± Y , ± i Y , ± Z , ± i Z } ≡ ⟨ X , Y , Z ⟩ {\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, G ≅ C 4 ∘ D 4 {\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. 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 σ 1 σ 2 σ 3 = i I {\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. Publications by Pauli eventually led to the eponym now in use. Basis elements of the algebra generate the Pauli group.
Multi-qubit Pauli group The Pauli group on n {\displaystyle n} qubits, G n {\displaystyle G_{n}} , is the group generated by the operators described above applied to each of n {\displaystyle n} qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle (\mathbb {C} ^{2})^{\otimes n}} . That is,
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