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. A graph state is a particular case of a stabilizer state as well as a 2-uniform hypergraph state, a generalization where the edges have cardinality between 1 and N.
Formal definition Quantum graph states can be defined in two equivalent ways: through the notion of quantum circuits and stabilizer formalism.
Quantum circuit definition Given a graph G = ( V , E ) {\displaystyle G=(V,E)} , with the set of vertices V {\displaystyle V} and the set of edges E {\displaystyle E} , the corresponding graph state is defined as
| G ⟩ = ∏ ( a , b ) ∈ E U { a , b } | + ⟩ ⊗ | V | {\displaystyle {\left|G\right\rangle }=\prod _{(a,b)\in E}U^{\{a,b\}}{\left|+\right\rangle }^{\otimes |V|}}
where | + ⟩ = 1 2 ( | 0 ⟩ + | 1 ⟩ ) {\displaystyle {\left|+\right\rangle }={\frac {1}{\sqrt {2}}}({\left|0\right\rangle }+{\left|1\right\rangle })} and the operator U { a , b } {\displaystyle U^{\{a,b\}}} is the controlled-Z interaction between the two vertices (corresponding to two qubits) a {\displaystyle a} and b {\displaystyle b}
U { a , b } = [ 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 − 1 ] {\displaystyle 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]}
Stabilizer formalism definition An alternative and equivalent definition is the following, which makes use of the stabilizer formalism. Define an operator S v {\displaystyle S_{v}} for each vertex v {\displaystyle v} of G {\displaystyle G} :
S v = σ x ( v ) ∏ u ∈ N ( v ) σ z ( u ) {\displaystyle S_{v}=\sigma _{x}^{(v)}\prod _{u\in N(v)}\sigma _{z}^{(u)}}
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