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.
where the operator is the controlled-Z interaction between the two vertices (qubits) a, b
An alternative and equivalent definition is the following.
Define an operator for each vertex a of G:
where N(a) is the neighborhood of a (that is, the set of all b such that ) and are the pauli matrices. The graph state is then defined as the simultaneous eigenstate of the operators with eigenvalue 1:
- M. Hein, J. Eisert, and H. J. Briegel (2004). "Multiparty entanglement in graph states". Physical Review A 69: 062311. doi:10.1103/PhysRevA.69.062311.
- S. Anders and H. J. Briegel (2006). "Fast simulation of stabilizer circuits using a graph-state representation". Physical Review A 73: 022334. doi:10.1103/PhysRevA.73.022334.
- Graph states on arxiv.org
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