Solving perfect matchings by frequency-grouped multi-photon events using a silicon chip

Computing the number of perfect matchings of a graph is a famous #P-complete problem. In this work, taking the advantages of the frequency dimension of photon, we propose and implement a photonic perfect matching solver, by combining two key techniques, frequency grouping and multi-photon counting. Based on a broadband photon-pair source from a silicon quantum chip and a wavelength-selective switch, we configure graphs up to sixteen vertices and estimate the perfect matchings of subgraphs up to six vertices. The experimental fidelities are more than 90% for all the graphs. Moreover, we demonstrate that the developed photonic system can enhance classical stochastic algorithms for solving nondeterministic-polynomial-time(NP) problems, such as the Boolean satisfiability problem and the densest subgraph. Our work contributes a promising method for solving the perfect matchings problem, which is simple in experiment setup and convenient to transform or scale up the object graph by regulating the frequency-correlated photon pairs.