Dimerized decomposition of quantum evolution on an arbitrary graph

  • He Feng
  • Tian-Min YanEmail author
  • Y. H. Jiang


The study of quantum evolution on graphs for diversified topologies is beneficial to modeling various realistic systems. A systematic method, the dimerized decomposition, is proposed to analyze the dynamics on an arbitrary network. By introducing global “flows” among interlinked dimerized subsystems, each of which locally consists of an input and an output port, the method provides an intuitive picture that the local properties of the subsystem are separated from the global structure of the network. The pictorial interpretation of quantum evolution as multiple flows through the graph allows for the analysis of the complex network dynamics supplementary to the conventional spectral method. Using the decomposition, the relation between spectral coefficients of adjacent sites with regard to individual dimer is obtained.


Continuous-time quantum walk Dimerized decomposition of Schrödinger equation Spectral analysis 



T.-M. Yan thanks M. Weidemüller for remarks and suggestions.


  1. 1.
    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915 (1998). ADSMathSciNetCrossRefGoogle Scholar
  2. 2.
    Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009). ADSMathSciNetCrossRefGoogle Scholar
  3. 3.
    Valkunas, L., Abramavicius, D., Mancal, T.: Molecular Excitation Dynamics and Relaxation: Quantum Theory and Spectroscopy, 1st edn. Wiley-VCH, Weinheim (2013)CrossRefGoogle Scholar
  4. 4.
    Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 022314 (2004). ADSCrossRefGoogle Scholar
  5. 5.
    Mohseni, M., Rebentrost, P., Lloyd, S., Aspuru-Guzik, A.: Environment-assisted quantum walks in photosynthetic energy transfer. J. Chem. Phys. 129(17), 174106 (2008). ADSCrossRefGoogle Scholar
  6. 6.
    Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91(20), 207901 (2003). ADSCrossRefGoogle Scholar
  7. 7.
    Mülken, O., Blumen, A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502(2–3), 37 (2011). ADSMathSciNetCrossRefGoogle Scholar
  8. 8.
    Côté, R., Russell, A., Eyler, E.E., Gould, P.L.: Quantum random walk with Rydberg atoms in an optical lattice. New J. Phys. 8(8), 156 (2006). ADSCrossRefGoogle Scholar
  9. 9.
    Mülken, O., Blumen, A., Amthor, T., Giese, C., Reetz-Lamour, M., Weidemüller, M.: Survival probabilities in coherent exciton transfer with trapping. Phys. Rev. Lett. 99(9), 090601 (2007). ADSCrossRefGoogle Scholar
  10. 10.
    Foulger, I., Gnutzmann, S., Tanner, G.: Quantum search on graphene lattices. Phys. Rev. Lett. 112(7), 070504 (2014). ADSCrossRefGoogle Scholar
  11. 11.
    Böhm, J., Bellec, M., Mortessagne, F., Kuhl, U., Barkhofen, S., Gehler, S., Stöckmann, H.J., Foulger, I., Gnutzmann, S., Tanner, G.: Microwave experiments simulating quantum search and directed transport in artificial graphene. Phys. Rev. Lett. 114(11), 110501 (2015). ADSCrossRefGoogle Scholar
  12. 12.
    Perets, H.B., Lahini, Y., Pozzi, F., Sorel, M., Morandotti, R., Silberberg, Y.: Realization of quantum walks with negligible decoherence in waveguide lattices. Phys. Rev. Lett. 100(17), 170506 (2008). ADSCrossRefGoogle Scholar
  13. 13.
    Aspuru-Guzik, A., Walther, P.: Photonic quantum simulators. Nat. Phys. 8(4), 285 (2012). CrossRefGoogle Scholar
  14. 14.
    Qiang, X., Loke, T., Montanaro, A., Aungskunsiri, K., Zhou, X., O’ Brien, J.L., Wang, J.B., Matthews, J.C.F.: Efficient quantum walk on a quantum processor. Nat. Commun. 7, 11511 (2016).
  15. 15.
    Novo, L., Chakraborty, S., Mohseni, M., Neven, H., Omar, Y.: Systematic dimensionality reduction for quantum walks: optimal spatial search and transport on non-regular graphs. Sci. Rep. 5, 13304 (2015). ADSCrossRefGoogle Scholar
  16. 16.
    Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 112(21), 210502 (2014). ADSCrossRefGoogle Scholar
  17. 17.
    Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett. 114(11), 110503 (2015). ADSCrossRefGoogle Scholar
  18. 18.
    Wong, T.G.: Diagrammatic approach to quantum search. Quantum Inf. Process. 14(6), 1767 (2015). ADSMathSciNetCrossRefzbMATHGoogle Scholar
  19. 19.
    Salimi, S.: Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quantum Inf. Process. 9(1), 75 (2010). MathSciNetCrossRefzbMATHGoogle Scholar
  20. 20.
    Koda, S.: Equivalence between a generalized dendritic network and a set of one-dimensional networks as a ground of linear dynamics. J. Chem. Phys. 142(20), 204112 (2015). ADSCrossRefGoogle Scholar
  21. 21.
    Sarkar, S., Kröber, D., Morr, D.K.: Equivalent resistance from the quantum to the classical transport limit. Phys. Rev. Lett. 117(22), 226601 (2016). ADSCrossRefGoogle Scholar
  22. 22.
    Wu, J., Tang, Z., Gong, Z., Cao, J., Mukamel, S.: Minimal model of quantum kinetic clusters for the energy-transfer network of a light-harvesting protein complex. J. Phys. Chem. Lett. 6(7), 1240 (2015). CrossRefGoogle Scholar
  23. 23.
    Cao, J., Silbey, R.J.: Optimization of exciton trapping in energy transfer processes. J. Phys. Chem. A 113(50), 13825 (2009). CrossRefGoogle Scholar
  24. 24.
    Burgarth, D., Maruyama, K., Nori, F.: Coupling strength estimation for spin chains despite restricted access. Phys. Rev. A 79(2), 020305 (2009). ADSCrossRefGoogle Scholar
  25. 25.
    Burgarth, D., Ajoy, A.: Evolution-free Hamiltonian parameter estimation through zeeman markers. Phys. Rev. Lett. 119(3), 030402 (2017). ADSCrossRefGoogle Scholar

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© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Shanghai Advanced Research InstituteChinese Academy of SciencesShanghaiChina
  2. 2.University of Chinese Academy of SciencesBeijingChina
  3. 3.ShanghaiTech UniversityShanghaiChina

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