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.

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 controlled-Z 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}}}

Alternative definition

An alternative and equivalent definition is the following.

Define an operator $K_{v}$ for each vertex v of G:

K_{v}=\sigma _{x}^{(v)}\prod _{u\in N(v)}\sigma _{z}^{(u)}

where $\sigma _{x,y,z}$ are the Pauli matrices and N(v) is the set of vertices adjacent to v. the $K_{v}$ operators commute. The graph state ${\left|G\right\rangle }$ is defined as the simultaneous $+1$ -eigenvalue eigenstate of the $\left|V\right|$ operators $\left\{K_{v}\right\}_{v\in V}$ :