- Graph states
In
quantum computing , is special type of multi-qubit state that can be represented by a graph. Each qubit is represented by a 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 ofentangled states.Graph states are useful in quantum error correcting codes, entanglement measurement and purification and for characterization of computational resources in measurement based quantum computing models.
Formal definition
Given a graph (G=(V,E), with the set of
vertices V and the set ofedges E, the corresponding graph state is defined as:left| G ight angle} =prod _{(a,b)in E}U^{{ a,b} } {left| + ight angle} ^{otimes V}
where the operator U^{{ a,b} } is the interaction between the two vertices (qubits) a, b
:U^{{ a,b} } =left [egin{array}{cccc} {1} & {0} & {0} & {0} \ {0} & {1} & {0} & {0} \ {0} & {0} & {1} & {0} \ {0} & {0} & {0} & {-1} end{array} ight]
And
:left| + ight angle} =fracleft| 0 ight angle} +{left| 1 ight angle} }{sqrt{2} } An alternative and equivalent definition is the following.
Define an operator K_{G}^{(a)} for each vertex a of G:
:K_{G}^{(a)} =sigma _{x}^{(a)} prod _{bin N(a)}sigma _{z}^{(b)}
Where N(a) is the neighborhood of a (that is, the set of all b such that a,b)in E) and sigma _{x,y,z} are the
pauli matrices . The graph state left| G ight angle} is then defined as the simultaneous eigenstate of the N=left|V ight| operators left{K_{G}^{(a)} ight}_{ain V} with eigenvalue 1::K_{G}^{(a)} {left| G ight angle} ={left| G ight angle}
See also
*
Entanglement
*cluster state References
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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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