Quantum Walk Search And Applications In Computation Science

Quantum walks is a newer and rapidly developing research field in quantum computing.In recent years,continuous-time(CTQW)and discrete-time(DTQW)quantum walks have been widely studied.Classical random walk,which is a very powerful tool for designing and developing random algorithms,is often used in engineering and scientific fields.Considering that quantum walk travels much further than their classical counterpart,so we have reasons to expect that quantum walks will have a superior performance in the design of quantum algorithms.In order to understand the important role of quantum walks in quantum computing better,this paper has done a more in-depth study on quantum walks.Our main work has two aspects:In this paper,we first discussed the notion of a continuous time quantum walk on a directed graph。The digraph with a non-hermitian adjacency matrix and the undirected graph with hermitian adjacency matrix are two major classifications in the graph theory.Starting from the properties of quantum walks in undirected graphs,we can investigate the properties of quantum on digraphs.In recent years,DTQW on a wide range of undirected graphs with a symmetric adjacency matrix has been extensively investigated,such as simulation of classical annealing processes,the novel properties of entanglement between correlated walkers.However,because quantum walks on a directed graph is non-unitary time-evolution,which is no guarantee of conservation of probability.And such walks,with non-unitary time evolution,is in-ability to experimentally implement using a quantum computer or quantum simulator.In this paper,we show that both of these obstacles can be overcome using unitary-dilation approach.For the few basic graphs that we consider,we show that the non-unitary continuous time evolution is related to the classical centrality of nodes in these basic graphs.Secondly,Considering the unitarity of the reconstructed matrix,the experimental steps of quantum walks on a digraph are designed.Whether the algorithm based on quantum walking can be used to analyze complex networks(usually composed of non-hermitian adjacency matrices)and provide effective sorting strategy has important theoretical and practical significance.In recently years,the progress in the fabrication of coupled optical waveguides lattices and the development of single photon sources has made it possible to experimentally investigate CTQW of single and correlated photons on discrete lattices.Noting that the node classical centrality in a digraph is correlated with the maximum node probability resulting from a continuoustime quantum walk.In this paper,the preparation of the initial state,the decomposition of6-dimensional unitary matrix and the realization of optical experiment are introduced by using linear optical devices.In order to reflects network centrality CTQW on digraphs,two three-node digraphs preparing the initial state on the global equal weight state are taken as an example,and then the unitary transformations are implemented.Finally,the experimental results proved that CTQW on digraphs is related to the classical centrality of node and PageRank.