An enhanced classical approach to graph isomorphism using continuous-time quantum walk

Abstract

Studies on graph isomorphism play an important role in graph research, and graph isomorphism algorithms have a wide range of applications in image matching, pattern recognition, computer vision, biochemistry and other fields. Previous research demonstrated that involving discrete-time quantum walk in the graph isomorphism algorithm could achieve complexity O(N 7 ) for general graphs, since quantum walk could be utilized as a new toolbox for solving graph problems. We develop an enhanced classical approach to graph isomorphism using continuous-time quantum walk, which has lower complexity O(N 5 ) and can effectively distinguish the graphs that are generally considered difficult. In addition, we define a graph similarity measure based on the proposed algorithm, which can be used for graph isomorphism and graph clustering. In the experiment, we test a wide variety of classes of graphs; the results show that the algorithm has a wide range of applications rather than being limited to a specific type of graph.

Bosonic Random Walk Neural Networks for Graph Learning

“…Quantum walks have also been shown to provide a model for universal computationChilds (2009). They have been explored for algorithmic applications Childs and Eisenberg (2003); Qiang et al (2012) and quantum simulation Berry et al (2005). While there have been multiple proposals of quantum neural networks over the years Gupta and Zia (2001); Biamonte et al (2017); there has been not much work done on quantum learning techniques for graphs.…”

Marking Vertices to Find Graph Isomorphism Mapping Based on Continuous-Time Quantum Walk

“…Thus we chose the Intuitive Method [24], Emms [21], and Qiang3 [24] for comparison. The Intuitive Method can discover an isomorphism mapping based on a judging algorithm (Qiang1) [23]. Emms [21,22] is most influential in the graph isomorphism problem, and we chose the continuous-time version [21].…”

“…Inspired by Douglas and Wang, Qiang proposed an enhanced approach called Qiang1 [23]. After performing CTQW on both graphs, it analyses the result and ascertains whether two graphs are isomorphic or not.…”

“…The algorithm named Qiang1 proposed by Qiang utilizes CTQW to ascertain whether two graphs are isomorphic [23]. It is accurate and fast, with a complexity of O(|V| 5 ) [23]. Nevertheless, it can only provide a judgement rather than a detailed isomorphism mapping.…”

Quantum walk neural networks with feature dependent coins

“…Quantum walks have recently received much attention because they have been shown to be a universal model for quantum computation (Childs 2009). In addition, they have numerous applications in quantum information science such as database search (Shenvi et al 2003), graph isomorphism (Qiang et al 2012), network analysis and navigation, and quantum simulation.…”