Analysis of Graph Structure Using Quantum Walks

Abstract

This thesis considers the use of the quantum walk for graph-based pattern recognition tasks. The main contributions are algorithms for: i) lifting cospectrality of graphs, ii) exact and inexact graph matching, iii) calculating graphs distances, and iv) embedding graphs. Since graphs are widely used to represent structured data, graph analysis algorithms form the basis of numerous solutions to pattern recognition problems. As such, good, error-tolerant graph analysis algorithms are a necessity. Random walks on graphs, both quantum and cla:ssical, are important tools because, while based on local connectivity structure, they capture global properties of the graphs being studied. While the use of the classical random walk for graph analysis has been studied, the use of the 'quantum walk has not. This thesis investigates the possibilities it offers. Chapter 2 of this thesis surveys the literature relating to graph analysis. Particular attention is paid to spectral and random walk based approaches. An overview of quantum computing and the qu~ntum walk is then given. In Chapter 3 we present a novel matrix representation for graphs. The spectrum of this II,latrix can be calculated in polynomial time and is able to distinguish between graphs for which there is no other sub-exponential time algorithm proven to be able to distinguish between them. 'rVe perform experiments on sets of strongly regular graphs, bipartite incidence graphs of balanced incomplete block designs and 3-level regular graphs. Such graphs form hard classes for any graph isomorphism algorithm. 'rVe also compare the performance of the spectrum of this new matrix representation with the performance of standard spectral methods. on large numbers of trees and regular graphs, graphs which are often encountered in real-world graph analysis problems. In all cases it significantly outperforms standard methods. Additionally, we use distances .... obtained from its spectrum to cluster graphs derived from real-world images. Thus, we provide a' spectr,al representation of graphs that can be used in place of standard spectral representations and which is far less prone to cospectrality. In Chapters 4 and 5 we present a graph structure that allows graph matching to be carried out by way of quantum interference between the walks on a pair of graphs of potentially different sizes. 'Interference amplitudes' between the walks are used as vertex-vertex assignment costs and the match is completed using the Hungarian algorithm. vVe also provide a method for further improving the match by rewarding edge-consistent assignments. vVe investigate the use of both discrete and continuous time quantum walks and carry out a comparative analysis. Using synthetic graphs with structural errors,' we show that it outperforms Umeyama's algorithm. We also propose a distance measure for graphs based on the interference amplitudes and show it can be used to successfully cluster Delaunay triangulations of images. In Chapter 6 we present an analysis of the commute time of the continuous-time quantum walk on a graph. For the classical random walk the commute time has been shown to be robust to errors in connectivity structure and to lead to spectral clustering algorithms with improved performance. vVe show how the commute time of the continuous-time quantum walk can be calculated and provide an analysis with reference to its classical counterpart. Experimentally, we show that the graph embedding that preserves the quantum commute-time is less prone to uni-dimensionality than the classical commute-time embedding and can be used to emphasise cluster structure. In Chapter 7 we present the conclusions that we draw from our work and areas which we believe merit further investigation. In summary, the work presented in this thesis demonstrates that quantum walks can be used to construct effective graph analysis algorithms. Furthermore, we have used such algorithms to provide solutions to problems where classical app'roaches have failed.EThOS - Electronic Theses Online ServiceGBUnited Kingdo