Graph Similarity Using Interfering Quantum Walks

We consider how continuous-time quantum walks can be used for graph matching, both exact and inexact, and measuring graph similarity. Our approach is to simulate the quantum walk on the two graphs in parallel by using an auxiliary graph that incorporates both graphs. The auxiliary graph allows quantum interference to take place between the two walks. Modelling the resultant interference amplitudes, which result from the differences in the two walks, we calculate probabilities for matches between pairs of vertices from the graphs. Using the Hungarian algorithm on these probabilities we recover a mapping between the graphs. To calculate graph similarity, we combine these probabilities with edge consistency information to give a consistency measure. We analyse our approach experimentally using synthetic graphs.