  We present a conceptually new 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. The new viewpoint immediately leads to many interesting results,
some of which we present here. Firstly, we identify a new and experimentally
completely unexplored multiphoton interference phenomenon. Secondly, 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. Thirdly, we explain how a
recent no-go result applies generally to linear optical quantum experiments,
thus revealing important insights to quantum state generation with current
photonic technology. Fourthly, 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 a novel perspective on a widely
used technology, and immediately raises many follow-up questions.
