Applications of combinatorial designs in computer science
ACM Computing Surveys (CSUR)
Algorithms for random generation and counting: a Markov chain approach
Algorithms for random generation and counting: a Markov chain approach
A fast quantum mechanical algorithm for database search
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Faster isomorphism testing of strongly regular graphs
STOC '96 Proceedings of the twenty-eighth annual ACM symposium on Theory of computing
Polynomial-Time Algorithms for Prime Factorization and Discrete Logarithms on a Quantum Computer
SIAM Journal on Computing
The anatomy of a large-scale hypertextual Web search engine
WWW7 Proceedings of the seventh international conference on World Wide Web 7
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
STOC '01 Proceedings of the thirty-third annual ACM symposium on Theory of computing
An Example of the Difference Between Quantum and Classical Random Walks
Quantum Information Processing
Edit Distance From Graph Spectra
ICCV '03 Proceedings of the Ninth IEEE International Conference on Computer Vision - Volume 2
Quantum Walk Algorithm for Element Distinctness
FOCS '04 Proceedings of the 45th Annual IEEE Symposium on Foundations of Computer Science
Exact and Approximate Graph Matching Using Random Walks
IEEE Transactions on Pattern Analysis and Machine Intelligence
Indexing Hierarchical Structures Using Graph Spectra
IEEE Transactions on Pattern Analysis and Machine Intelligence
Polynomial-Time Metrics for Attributed Trees
IEEE Transactions on Pattern Analysis and Machine Intelligence
On the structure of the adjacency matrix of the line digraph of a regular digraph
Discrete Applied Mathematics
On the relation between quantum walks and zeta functions
Quantum Information Processing
Pattern analysis with graphs: Parallel work at Bern and York
Pattern Recognition Letters
Graph characterization using gaussian wave packet signature
SIMBAD'13 Proceedings of the Second international conference on Similarity-Based Pattern Recognition
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In this paper we explore how a spectral technique suggested by coined quantum walks can be used to distinguish between graphs that are cospectral with respect to standard matrix representations. The algorithm runs in polynomial time and, moreover, can distinguish many graphs for which there is no subexponential time algorithm that is proven to be able to distinguish between them. In the paper, we give a description of the coined quantum walk from the field of quantum computing. The evolution of the walk is governed by a unitary matrix. We show how the spectrum of this matrix is related to the spectrum of the transition matrix of the classical random walk. However, despite this relationship the behaviour of the quantum walk is vastly different from the classical walk. This leads us to define a new matrix based on the amplitudes of paths of the walk whose spectrum we use to characterise graphs. We carry out three sets of experiments using this matrix representation. Firstly, we test the ability of the spectrum to distinguish between sets of graphs that are cospectral with respect to standard matrix representation. These include strongly regular graphs, and incidence graphs of balanced incomplete block designs (BIBDs). Secondly, we test our method on ALL regular graphs on up to 14 vertices and ALL trees on up to 24 vertices. This demonstrates that the problem of cospectrality is often encountered with conventional algorithms and tests the ability of our method to resolve this problem. Thirdly, we use distances obtained from the spectra of S^+(U^3) to cluster graphs derived from real-world image data and these are qualitatively better than those obtained with the spectra of the adjacency matrix. Thus, we provide a spectral representation of graphs that can be used in place of standard spectral representations, far less prone to the problems of cospectrality.