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| In [[quantum computing]], a '''graph state''' is a special type of multi-[[qubit]] state that can be represented by a [[graph (mathematics)|graph]]. Each qubit is represented by a [[Vertex (graph theory)|vertex]] of the graph, and there is an edge between every interacting pair of qubits. In particular, they are a convenient way of representing certain types of [[Entanglement (graph measure)|entangled]] states.
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| Graph states are useful in [[quantum error-correcting code]]s, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models.
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| == Formal definition ==
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| Given a graph ''G'' = (''V'', ''E''), with the set of [[vertex (graph theory)|vertices]] ''V'' and the set of [[Glossary of graph theory#Basics|edges]] ''E'', the corresponding graph state is defined as
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| :<math>{\left| G \right\rangle} =\prod _{(a,b)\in E}U^{\{ a,b\} } {\left| + \right\rangle} ^{\otimes V}</math>
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| where the operator <math>U^{\{ a,b\} }</math> is the [[Quantum_gate#Controlled_gates|controlled-''Z'']] interaction between the two vertices (qubits) ''a'', ''b''
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| :<math> U^{\{ a,b\} } =\left[\begin{array}{cccc} {1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {-1} \end{array}\right]</math>
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| And
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| :<math>{\left| + \right\rangle} =\frac{{\left| 0 \right\rangle} +{\left| 1 \right\rangle} }{\sqrt{2} } </math>
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| === Alternative definition ===
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| An alternative and equivalent definition is the following.
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| Define an operator <math>K_{G}^{(a)}</math> for each vertex ''a'' of ''G'':
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| :<math>K_{G}^{(a)} =\sigma _{x}^{(a)} \prod _{b\in N(a)}\sigma _{z}^{(b)} </math>
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| where ''N''(''a'') is the neighborhood of ''a'' (that is, the set of all ''b'' such that <math>(a,b)\in E</math>) and <math> \sigma _{x,y,z}</math> are the [[pauli matrices]]. The graph state <math>{\left| G \right\rangle}</math> is then defined as the simultaneous eigenstate of the <math>N=\left|V\right|</math> operators <math> \left\{K_{G}^{(a)} \right\}_{a\in V} </math> with eigenvalue 1:
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| :<math>K_{G}^{(a)} {\left| G \right\rangle} ={\left| G \right\rangle} </math>
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| == See also ==
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| * [[Entanglement (graph measure)|Entanglement]]
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| * [[Cluster state]]
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| ==References==
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| * {{cite journal | author=M. Hein, J. Eisert, and H. J. Briegel | title=Multiparty entanglement in graph states| journal=[[Physical Review A]] | year=2004| volume=69 | pages=062311 | doi=10.1103/PhysRevA.69.062311}}
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| * {{cite journal | author=S. Anders and H. J. Briegel | title=Fast simulation of stabilizer circuits using a graph-state representation| journal=[[Physical Review A]] | year=2006| volume=73 | pages=022334 | doi=10.1103/PhysRevA.73.022334 }}
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| *[http://xstructure.inr.ac.ru/x-bin/theme3.py?level=1&index1=423009 Graph states on arxiv.org]
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| [[Category:Quantum information science]]
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| {{Comp-sci-stub}}
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