Quantum experiments and graphs II: Quantum interference, computation, and state generation

Significance Graph theory can be used to model and explain different phenomena from physics. In this paper, we show that one can interpret quantum experiments composed of linear optics and probabilistic sources with graph theory. The important understanding is that complex weights in the graph naturally describe the quantum interference. That point of view leads to several results: We uncover a yet unexplored type of multiphoton quantum interference. It can be used to solve questions that are intractable on a classical computer. Also, the connection points toward a general restriction on creating certain classes of quantum states. In general, it gives another perspective on a mature photonic technology and will be significant for future designs of such experiments. We present an approach to describe state-of-the-art photonic quantum experiments using graph theory. There, the quantum states are given by the coherent superpositions of perfect matchings. The crucial observation is that introducing complex weights in graphs naturally leads to quantum interference. This viewpoint immediately leads to many interesting results, some of which we present here. First, we identify an experimental unexplored multiphoton interference phenomenon. Second, we find that computing the results of such experiments is #P-hard, which means it is a classically intractable problem dealing with the computation of a matrix function Permanent and its generalization Hafnian. Third, we explain how a recent no-go result applies generally to linear optical quantum experiments, thus revealing important insights into quantum state generation with current photonic technology. Fourth, we show how to describe quantum protocols such as entanglement swapping in a graphical way. The uncovered bridge between quantum experiments and graph theory offers another perspective on a widely used technology and immediately raises many follow-up questions.