Graph states

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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 of entangled 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.

[edit] Formal definition

Given a graph (G=(V,E), with the set of vertices V and the set of edges E, the corresponding graph state is defined as

${\left| G \right\rangle} =\prod _{(a,b)\in E}U^{\{ a,b\} } {\left| + \right\rangle} ^{\otimes V}$

where the operator $U {a, b}$ is the interaction between the two vertices (qubits) a, b

$U^{\{ a,b\} } =\left[\begin{array}{cccc} {1} & {0} & {0} & {0} \\ {0} & {1} & {0} & {0} \\ {0} & {0} & {1} & {0} \\ {0} & {0} & {0} & {-1} \end{array}\right]$

And

${\left| + \right\rangle} =\frac{{\left| 0 \right\rangle} +{\left| 1 \right\rangle} }{\sqrt{2} }$

An alternative and equivalent definition is the following.

Define an operator $K_{G}^{(a)}$ for each vertex a of G:

$K_{G}^{(a)} =\sigma _{x}^{(a)} \prod _{b\in N(a)}\sigma _{z}^{(b)}$

Where N(a) is the neighborhood of a (that is, the set of all b such that $(a,b)\in E$ ) and $σ x, y, z$ are the pauli matrices. The graph state ${\left| G \right\rangle}$ is then defined as the simultaneous eigenstate of the $N=\left|V\right|$ operators $\left\{K_{G}^{(a)} \right\}_{a\in V}$ with eigenvalue 1: