# Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings. (arXiv:1705.06646v2 [quant-ph] UPDATED)

We show a surprising link between experimental setups to realize

high-dimensional multipartite quantum states and Graph Theory. In these setups,

the paths of photons are identified such that the photon-source information is

never created. We find that each of these setups corresponds to an undirected

graph, and every undirected graph corresponds to an experimental setup. Every

term in the emerging quantum superposition corresponds to a perfect matching in

the graph. Calculating the final quantum state is in the complexity class

#P-complete, thus cannot be done efficiently. To strengthen the link further,

theorems from Graph Theory -- such as Hall's marriage problem -- are rephrased

in the language of pair creation in quantum experiments. We show explicitly how

this link allows to answer questions about quantum experiments (such as which

classes of entangled states can be created) with graph theoretical methods, and

potentially simulate properties of Graphs and Networks with quantum experiments

(such as critical exponents and phase transitions).