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[[File:Cayley_graph_Pauli.svg|thumb|The [[Möbius–Kantor graph]], the [[Cayley graph]] of the Pauli group <math>G_1</math> with generators ''X'', ''Y'', and ''Z'']] | |||
In [[physics]] and [[mathematics]], the '''Pauli group''' <math>G_1</math> on 1 [[qubit]] is the 16-element [[matrix group]] consisting of the 2 × 2 [[identity matrix]] <math>I</math> and all of the [[Pauli matrices]] | |||
:<math>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}</math>, | |||
together with the products of these matrices with the factors <math>-1</math> and <math>\pm i</math>: | |||
:<math>G_1 \ \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</math>. | |||
The Pauli group is [[Generating_set_of_a_group|generated]] by the Pauli matrices, and like them it is named after [[Wolfgang Pauli]]. | |||
The Pauli group on n qubits, <math>G_n</math>, is the group generated by the operators described above applied to each of <math>n</math> qubits in the [[tensor product]] [[Hilbert space]] <math>(\mathbb{C}^2)^{\otimes n}</math>. | |||
==References== | |||
* {{cite book |title= Quantum Computation and Quantum Information|last= Nielsen|first= Michael A|authorlink= |coauthors= Chuang, Isaac L|year= 2000|publisher= [[Cambridge University Press]]|location= [[Cambridge]]; [[New York City|New York]]|isbn= 978-0-521-63235-5|oclc= 43641333|pages= }} | |||
[[Category:Finite groups]] | |||
[[Category:Quantum information science]] | |||
{{quantum-stub}} |
Revision as of 23:54, 11 April 2013
In physics and mathematics, the Pauli group on 1 qubit is the 16-element matrix group consisting of the 2 × 2 identity matrix and all of the Pauli matrices
together with the products of these matrices with the factors and :
The Pauli group is generated by the Pauli matrices, and like them it is named after Wolfgang Pauli.
The Pauli group on n qubits, , is the group generated by the operators described above applied to each of qubits in the tensor product Hilbert space .
References
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