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

## TL;DR

This paper establishes a novel connection between high-dimensional multipartite quantum states and graph theory, showing how experimental setups correspond to graphs and superpositions relate to perfect matchings, with implications for quantum state creation and simulation.

## Contribution

It introduces a framework linking quantum experiments to graph theory, enabling analysis of entangled states and properties of graphs through quantum setups.

## Key findings

- Quantum states correspond to perfect matchings in graphs
- Calculating the quantum state is #P-complete
- Graph theoretical methods can analyze quantum experiment capabilities

## Abstract

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).