Abstract
In this paper we explore how a spectral technique suggested by quantum walks can be used to distinguish non-isomorphic cospectral graphs. Reviewing ideas from the field of quantum computing we recall the definition of the unitary matrices inducing quantum walks. We show how the spectra of these matrices are related to the spectra of the transition matrices of classical walks. Despite this relationship the behaviour of quantum walks is vastly different from classical walks. We show how this leads us to define a new matrix whose spectrum can be used to distinguish between graphs that are otherwise indistinguishable by standard spectral methods.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
Lovász, L.: Random Walks on Graphs: A Survey. In: Combinatorics, Paul Erdös is Eighty, vol. 2, pp. 353–398. János Bolyai Mathematical Society, Budapest (1996)
Sinclair, A.: Algorithms for random generation and counting: a Markov chain approach. Birkhauser, Boston (1993)
Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30, 107–117 (1998)
Robles-Kelly, A., Hancock, E.R.: String edit distance, random walks and graph matching. International Journal of Pattern Recognition and Artificial Intelligence 18, 315–327 (2004)
Robles-Kelly, A., Hancock, E.R.: Edit distance from graph spectra. In: Proc. of the IEEE International Conference on Computer Vision, pp. 234–241 (2003)
Gori, M., Maggini, M., Sarti, L.: Graph matching using random walks. In: IEEE 17th International Conference on Pattern Recognition (2004)
Cameron, P.J.: Strongly regular graphs. In: Topics in Algebraic Graph Theory, pp. 203–221. Cambridge University Press, Cambridge (2004)
Schwenk, A.J.: Almost all trees are cospectral. In: Harary, F. (ed.) New Directions in the Theory of Graphs, pp. 275–307. Academic Press, London (1973)
Merris, R.: Almost all trees are coimmanantal. Linear Algebra and its Applications 150, 61–66 (1991)
Kempe, J.: Quantum random walks – an introductory overview. Contemporary Physics 44, 307–327 (2003)
Ambainis, A.: Quantum walks and their algorithmic applications. International Journal of Quantum Information 1, 507–518 (2003)
Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67 (2003)
Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Information Processing 1, 35 (2002)
Kempe, J.: Quantum random walks hit exponentially faster. In: Arora, S., Jansen, K., Rolim, J.D.P., Sahai, A. (eds.) RANDOM 2003 and APPROX 2003. LNCS, vol. 2764, pp. 354–369. Springer, Heidelberg (2003)
Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proc. of 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 (2004)
Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proc. 33th STOC, pp. 50–59. ACM Press, New York (2001)
Rudolph, T.: Constructing physically intuitive graph invariants (2002)
Severini, S.: On the digraph of a unitary matrix. SIAM Journal on Matrix Analysis and Applications 25, 295–300 (2003)
Spence, E.: Strongly Regular Graphs (2004), http://www.maths.gla.ac.uk/es/srgraphs.htm
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer-Verlag Berlin Heidelberg
About this paper
Cite this paper
Emms, D., Severini, S., Wilson, R.C., Hancock, E.R. (2005). Towards Unitary Representations for Graph Matching. In: Brun, L., Vento, M. (eds) Graph-Based Representations in Pattern Recognition. GbRPR 2005. Lecture Notes in Computer Science, vol 3434. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-540-31988-7_14
Download citation
DOI: https://doi.org/10.1007/978-3-540-31988-7_14
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-25270-2
Online ISBN: 978-3-540-31988-7
eBook Packages: Computer ScienceComputer Science (R0)