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We show a surprising link between experimental setups to realizehigh-dimensional multipartite quantum states and Graph Theory. In these setups,the paths of photons are identified such that the photon-source information isnever created. We find that each of these setups corresponds to an undirectedgraph, and every undirected graph corresponds to an experimental setup. Everyterm in the emerging quantum superposition corresponds to a perfect matching inthe 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 rephrasedin the language of pair creation in quantum experiments. We show explicitly howthis link allows to answer questions about quantum experiments (such as whichclasses of entangled states can be created) with graph theoretical methods, andpotentially simulate properties of Graphs and Networks with quantum experiments(such as critical exponents and phase transitions).

Quantum Experiments and Graphs: Multiparty States as coherent superpositions of Perfect Matchings