Abstract
In this paper we consider the problem of distinguishing graphs that are cospectral with respect to the standard adjacency and Laplacian matrix representations. Borrowing ideas from the field of quantum computing, we define a new matrix based on paths of the coined quantum walk. Quantum walks exhibit interference effects and their behaviour is markedly different to that of classical random walks. We show that the spectrum of this new matrix is able to distinguish many graphs which cannot be distinguished by standard spectral methods. We pay particular attention to strongly regular graphs; if a pair of strongly regular graphs share the same parameter set then there is no efficient algorithm that is proven to be able distinguish them. We have tested the method on large families of co-parametric strongly regular graphs and found it to be successful in every case. We have also tested the spectra’s performance when used to give a distance measure for inexact graph matching tasks.
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Aharonov, D., Ambainis, A., Kempe, J., Vazirani, U.: Quantum walks on graphs. In: Proc. 33th STOC, pp. 50–59. ACM, New York (2001)
Aldous, D., Fill, J.: Reversible markov chains and random walks on graphs (2005)
Ambainis, A.: Quantum walks and their algorithmic applications. International Journal of Quantum Information 1, 507–518 (2003)
Ambainis, A.: Quantum walk algorithm for element distinctness. In: Proc. of 45th Annual IEEE Symposium on Foundations of Computer Science, FOCS 2004 (2004)
Ambainis, A., Bach, E., Nayak, A., Vishwanath, A., Watrous, J.: One-dimensional quantum walks. In: Proc. 33th STOC, pp. 60–69. ACM, New York (2001)
Braunstein, S.L.: Quantum teleportation without irreversible detection. Physical Review A, 1900–1903 (1996)
Brin, S., Page, L.: The anatomy of a large-scale hypertextual Web search engine. Computer Networks and ISDN Systems 30(1-7), 107–117 (1998)
Cameron, P.J.: Topics in Algebraic Graph Theory. In: Chapter Strongly regular graphs, pp. 203–221. Cambridge University Press, Cambridge (2004)
Childs, A., Farhi, E., Gutmann, S.: An example of the difference between quantum and classical random walks. Quantum Information Processing 1, 35 (2002)
Cirac, J., Zoller, P., Kimble, H.J., Mabuchi, H.: Quantum state transfer and entanglement distribution among distant nodes in a quantum network. Phys. Rev. Lett. 78, 3221–3224 (1997)
Gori, M., Maggini, M., Sarti, L.: Graph matching using random walks. In: IEEE 17th International Conference on Pattern Recognition (August 2004)
Grover, L.: A fast quantum mechanical algorithm for database search. In: Proc. 28th Annual ACM Symposium on the Theory of Computation, pp. 212–219. ACM Press, New York (1996)
Kempe, J.: Quantum random walks – an introductory overview. Contemporary Physics 44(4), 307–327 (2003)
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)
Kwiat, P., Mitchell, J.R., Schwindt, P.D.D., White, A.G.
Lovász, L.: Random Walks on Graphs: A Survey. In: Paul Erdös is Eighty, vol. 2, pp. 353–398. János Bolyai Mathematical Society, Budapest (1996)
Merris, R.: Almost all trees are coimmanantal. Linear Algebra and its Applications 150, 61–66 (1991)
Nayak, A., Vishwanath, A.: Quantum walk on a line, DIMACS Technical Report 2000-43 (2000)
Nielson, M.A., Chuang, I.L.: Quantum Computing and Quantum Information. Cambridge University Press, Cambridge (2000)
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)
Robles-Kelly, A., Hancock, E.R.: String edit distance, random walks and graph matching. International Journal of Pattern Recognition and Artificial Intelligence 18(3), 315–327 (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)
Severini, S.: On the digraph of a unitary matrix. SIAM Journal on Matrix Analysis and Applications 25(1), 295–300 (2003)
Shenvi, N., Kempe, J., Whaley, K.B.: A quantum random walk search algorithm. Phys. Rev. A 67(5) (2003)
Sinclair, A.: Algorithms for random generation and counting: a Markov chain approach. Birkhauser, Boston (1993)
Spence, E.: Strongly Regular Graphs (2004), http://www.maths.gla.ac.uk/es/srgraphs.htm
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Emms, D., Severini, S., Wilson, R.C., Hancock, E.R. (2005). Coined Quantum Walks Lift the Cospectrality of Graphs and Trees. In: Rangarajan, A., Vemuri, B., Yuille, A.L. (eds) Energy Minimization Methods in Computer Vision and Pattern Recognition. EMMCVPR 2005. Lecture Notes in Computer Science, vol 3757. Springer, Berlin, Heidelberg. https://doi.org/10.1007/11585978_22
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DOI: https://doi.org/10.1007/11585978_22
Publisher Name: Springer, Berlin, Heidelberg
Print ISBN: 978-3-540-30287-2
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