Abstract
We study the dynamics of continuous-time quantum walks (CTQW) on networks with highly degenerate eigenvalue spectra of the corresponding connectivity matrices. In particular, we consider the two cases of a star graph and of a complete graph, both having one highly degenerate eigenvalue, while displaying different topologies. While the CTQW spreading over the network—in terms of the average probability to return or to stay at an initially excited node—is in both cases very slow, also when compared to the corresponding classical continuous-time random walk (CTRW), we show how the spreading is enhanced by randomly adding bonds to the star graph or removing bonds from the complete graph. Then, the spreading of the excitations may become very fast, even outperforming the corresponding CTRW. Our numerical results suggest that the maximal spreading is reached halfway between the star graph and the complete graph. We further show how this disorder-enhanced spreading is related to the networks’ eigenvalues.
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References
Alexander S., Orbach R.: Density of states on fractals: “fractons”. J. Phys. (Paris) Lett. 43, 625–631 (1982)
Bray A., Rodgers G.: Diffusion in a sparsely connected space: a model for glassy relaxation. Phys. Rev. B 38, 11461 (1988)
Burgarth D., Maruyama K., Nori F.: Coupling strength estimation for spin chains despite restricted access. Phys. Rev. A 79, 020305 (2009)
Cvetković D., Doob M., Sachs H.: Spectra of Graphs: Theory and Applications. Academic Press, New York, NY (1997)
Darázs Z., Kiss T.: Pólya number of the continuous-time quantum walks. Phys. Rev. A 81, 062319 (2010)
Doi M., Edwards S.: The Theory of Polymer Dynamics. Oxford University Press, Oxford (1988)
Farhi E., Gutmann S.: Quantum computation and decision trees. Phys. Rev. A 58, 915 (1998)
Feldman E., Hillery M., Lee H.W., Reitzner D., Zheng H., Bužek V.: Quantum computation and decision trees. Phys. Rev. A 82, 040301 (2010)
Kempe J.: Quantum random walks—an introductory overview. Contemp. Phys. 44, 307 (2003)
Kenkre V., Reineker P.: Exciton Dynamics in Molecular Crystals and Aggregates. Springer, Berlin (1982)
Mülken O., Bierbaum V., Blumen A.: Coherent exciton transport in dendrimers and continuous-time quantum walks. J. Chem. Phys. 124, 124905 (2006)
Mülken O., Blumen A.: Spacetime structures of continuous-time quantum walks. Phys. Rev. E 71, 036128 (2005)
Mülken O., Blumen A.: Efficiency of quantum and classical transport on graphs. Phys. Rev. E 73, 066117 (2006)
Mülken O., Blumen A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502, 37 (2011)
Mülken O., Pernice V., Blumen A.: Quantum transport on small-world networks: a continuous-time quantum walk approach. Phys. Rev. E 76, 051125 (2007)
Mülken O., Volta A., Blumen A.: Asymmetries in symmetric quantum walks on two-dimensional networks. Phys. Rev. A 72, 042334 (2005)
Reitzner D., Hillery M., Feldman E., Bužek V.: Quantum searches on highly symmetric graphs. Phys. Rev. A 79, 012323 (2009)
Salimi S.: Continuous-time quantum walks on star graphs. Ann. Phys. 324, 1185–1193 (2009)
van Kampen N.: Stochastic Processes in Physics and Chemistry. North Holland, Amsterdam (1992)
Xu X.: Exact analytical results for quantum walks on star graphs. J. Phys. A 42, 115205 (2009)
Ziman J.: Principles of the Theory of Solids. Cambridge University Press, Cambridge, UK (1979)
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Anishchenko, A., Blumen, A. & Mülken, O. Enhancing the spreading of quantum walks on star graphs by additional bonds. Quantum Inf Process 11, 1273–1286 (2012). https://doi.org/10.1007/s11128-012-0376-9
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DOI: https://doi.org/10.1007/s11128-012-0376-9