The Research Of Graph Isomorphism Algorithm Based On Quantum Walk

Graph matching has a wide range of applications in computer vision, pattern recognition, information retrieval, artificial intelligence, biochemistry and other fields. Graph isomorphism is the kernel of graph matching, and the approach to graph isomorphism lays the foundation for graph matching. Moreover, the computational complexity of graph isomorphism makes it considerable theoretical important in mathematics. Up to now, there is no graph isomorphism algorithm with polynomial complexity for general graphs in the classical computation. As a new model of quantum computation, quantum walk provides new ideas for graph isomorphism. There have been several algorithms based on quantum walk, which show significant improvements compared with the classical algorithms.Based on the researches about the models of quantum walk, this paper carries out researches about graph isomorphism problem in the following four aspects. Firstly, classifying and analyzing the proposed graph isomorphism algorithms based on quantum walk, we discuss the shorts of the algorithms and propose the ways for improvements. Meanwhile, we analyze the ways in which quantum walk models are used in the algorithms.Secondly, based on continuous-time quantum walk, we propose an algorithm for distinguishing non-isomorphic graphs. The algorithm has the lowest computational copofr omspipmolseielxa airt yng e rOwap( hm se5 aa)rs eua rstee f soatfre dga.rs a Tkphnheo swriem saiunlladtrs i dtsyihs botiawnsg etudhi est hhceeos ar rtlhegceot rnsitiehmsmsi l.aa nIrn dg trwhaepid heesx paepeffrpeilmcicteiavnbetislly,i t.ay W looeft the algorithm.Thirdly, based on the above algorithm for distinguishing graphs, we propose an algorithm for finding the isomorphism map between the isomorphic graphs. The atchlogeno n reeixtchptimeor nihm paerso npbtsoa,lb y iwnlioteim e sti eabsle tc t wovmeaerpineu ttviaeetsrio t incoaefl s .cr Taonmhdepo almelgx oigtryrit a hpOmh(s h a6ws) i h tahing dhd ipwfrfioedrbeea nbatpi l pivtlyeic roatifbc ieflisint y da.i nnIdgn the correct isomorphism map, which shows the effectiveness of algorithm.Finally, based on the comparison of probability amplitudes, we propose an algorithm for finding the isomorphism map between the isomorphic graphs. The atthelgseto elrxoitathscm to ifhs oraamsn otdhroepm hl iosgwmrae psmth ascp ow mbietphtuw tdaeitefifnoe nrreaanln tdc vooemmrtp iglcereaxspi thayns dOe fc(f oe cn4tni)ve ceatlsiyo .f naI nrp arthos ebk aenbxoiplwiet.ir eiImst cbeanenttws ,f eiwnedne vertices. The results verify the effectiveness of algorithm, and show the wide applicability of the algorithm.