Pauli group (original) (raw)

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

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

The Pauli group consists of the 2 × 2 identity matrix I {\displaystyle I} $I$ 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}}} $X=\sigma _{1}={\begin{pmatrix}0&1\\1&0\end{pmatrix}},\quad Y=\sigma _{2}={\begin{pmatrix}0&-i\\i&0\end{pmatrix}},\quad Z=\sigma _{3}={\begin{pmatrix}1&0\\0&-1\end{pmatrix}}$ ,

together with the products of these matrices with the factors ± 1 {\displaystyle \pm 1} $\pm 1$ and ± i {\displaystyle \pm i} $\pm i$ :

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

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}} $G\ \cong C_{4}\circ D_{4}$ 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 σ 1 σ 2 σ 3 = i I {\displaystyle \sigma _{1}\sigma _{2}\sigma _{3}=iI} $\sigma _{1}\sigma _{2}\sigma _{3}=iI$ whereas there is no such relationship for the gamma group.

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 is based on qubits. The Pauli group on n {\displaystyle n} $n$ qubits, G n {\displaystyle G_{n}} $G_{n}$ , is the group generated by the operators described above applied to each of n {\displaystyle n} $n$ qubits in the tensor product Hilbert space ( C 2 ) ⊗ n {\displaystyle (\mathbb {C} ^{2})^{\otimes n}} $(\mathbb {C} ^{2})^{\otimes n}$ . That is,

G n = ⟨ W 1 ⊗ ⋯ ⊗ W n : W i ∈ { I , X , Y , Z } ⟩ . {\displaystyle G_{n}=\langle W_{1}\otimes \cdots \otimes W_{n}:W_{i}\in \{I,X,Y,Z\}\rangle .} $G_{n}=\langle W_{1}\otimes \cdots \otimes W_{n}:W_{i}\in \{I,X,Y,Z\}\rangle .$

The order of G n {\displaystyle G_{n}} $G_{n}$ is 4 ⋅ 4 n {\displaystyle 4\cdot 4^{n}} $4\cdot 4^{n}$ since a scalar ± 1 {\displaystyle \pm 1} $\pm 1$ or ± i {\displaystyle \pm i} $\pm i$ factor in any tensor position can be moved to any other position.

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.

^ Pauli group on GroupNames
^ L. E. Dickson (1914) Linear Algebras, pages 13,4

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