Graph state

In quantum computing, a graph state is a 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.

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

where the operator ${\displaystyle U^{\{a,b\}}}$ is the controlled-Z interaction between the two vertices (qubits) a, b

And

An alternative and equivalent definition is the following.

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

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