Alternated two-particle discrete-time quantum walk on arbitrary graphs and its application on graph isomorphism testing

Abstract This paper introduces a novel alternated two-particle discrete-time quantum walk model on arbitrary graphs, overcoming the limitation that conventional shared coin schemes are confined to regular graphs. By employing a fixed-dimension coin operator and encoding the graph structural information into the shift operator, the new model becomes applicable to arbitrary graphs. Further, by adding interaction between the two particles, an extension is presented, which can enhance the model’s dynamic properties and facilitate more intricate quantum interference phenomena. Based on the new model, a kind of quantum graph isomorphism algorithm framework is proposed. It just need O ( N 0.5 *∣ E ∣) steps of quantum walk and O ( N 2 ) dimensions of Hilbert space, which offers a significant reduction in complexity compared to other quantum walk based algorithms. The graph isomorphism testing is performed in the framework using non-interacting and interacting quantum walks respectively. Experimental results illustrate that the algorithm based on interacting particles achieves a 100% success rate in discriminating all test graphs. While the algorithm with non-interacting particles performs a 100% distinction success rate on general non-regular graphs, albeit being ineffective on strongly regular graphs. Due to the effectiveness of the algorithm on arbitrary graphs, the algorithm has broad application prospects in the identification and comparison of chemical molecular structures, the analysis of social networks and so on.