Skip to main content
SpringerLink
Log in

Dimerized decomposition of quantum evolution on an arbitrary graph

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

References

  1. Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58(2), 915 (1998). https://doi.org/10.1103/PhysRevA.58.915

    Article  ADS  MathSciNet  Google Scholar 

  2. Childs, A.M.: Universal computation by quantum walk. Phys. Rev. Lett. 102(18), 180501 (2009). https://doi.org/10.1103/PhysRevLett.102.180501

    Article  ADS  MathSciNet  Google Scholar 

  3. Valkunas, L., Abramavicius, D., Mancal, T.: Molecular Excitation Dynamics and Relaxation: Quantum Theory and Spectroscopy, 1st edn. Wiley-VCH, Weinheim (2013)

    Book  Google Scholar 

  4. Childs, A.M., Goldstone, J.: Spatial search by quantum walk. Phys. Rev. A 70(2), 022314 (2004). https://doi.org/10.1103/PhysRevA.70.022314

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1063/1.3002335

    Article  ADS  Google Scholar 

  6. Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91(20), 207901 (2003). https://doi.org/10.1103/PhysRevLett.91.207901

    Article  ADS  Google Scholar 

  7. Mülken, O., Blumen, A.: Continuous-time quantum walks: models for coherent transport on complex networks. Phys. Rep. 502(2–3), 37 (2011). https://doi.org/10.1016/j.physrep.2011.01.002

    Article  ADS  MathSciNet  Google Scholar 

  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). https://doi.org/10.1088/1367-2630/8/8/156

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.99.090601

    Article  ADS  Google Scholar 

  10. Foulger, I., Gnutzmann, S., Tanner, G.: Quantum search on graphene lattices. Phys. Rev. Lett. 112(7), 070504 (2014). https://doi.org/10.1103/PhysRevLett.112.070504

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.114.110501

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.100.170506

    Article  ADS  Google Scholar 

  13. Aspuru-Guzik, A., Walther, P.: Photonic quantum simulators. Nat. Phys. 8(4), 285 (2012). https://doi.org/10.1038/nphys2253

    Article  Google Scholar 

  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). https://doi.org/10.1038/ncomms11511

  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). https://doi.org/10.1038/srep13304

    Article  ADS  Google Scholar 

  16. Janmark, J., Meyer, D.A., Wong, T.G.: Global symmetry is unnecessary for fast quantum search. Phys. Rev. Lett. 112(21), 210502 (2014). https://doi.org/10.1103/PhysRevLett.112.210502

    Article  ADS  Google Scholar 

  17. Meyer, D.A., Wong, T.G.: Connectivity is a poor indicator of fast quantum search. Phys. Rev. Lett. 114(11), 110503 (2015). https://doi.org/10.1103/PhysRevLett.114.110503

    Article  ADS  Google Scholar 

  18. Wong, T.G.: Diagrammatic approach to quantum search. Quantum Inf. Process. 14(6), 1767 (2015). https://doi.org/10.1007/s11128-015-0959-3

    Article  ADS  MathSciNet  MATH  Google Scholar 

  19. Salimi, S.: Continuous-time quantum walks on semi-regular spidernet graphs via quantum probability theory. Quantum Inf. Process. 9(1), 75 (2010). https://doi.org/10.1007/s11128-009-0130-0

    Article  MathSciNet  MATH  Google Scholar 

  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). https://doi.org/10.1063/1.4921730

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1103/PhysRevLett.117.226601

    Article  ADS  Google Scholar 

  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). https://doi.org/10.1021/acs.jpclett.5b00227

    Article  Google Scholar 

  23. Cao, J., Silbey, R.J.: Optimization of exciton trapping in energy transfer processes. J. Phys. Chem. A 113(50), 13825 (2009). https://doi.org/10.1021/jp9032589

    Article  Google Scholar 

  24. Burgarth, D., Maruyama, K., Nori, F.: Coupling strength estimation for spin chains despite restricted access. Phys. Rev. A 79(2), 020305 (2009). https://doi.org/10.1103/PhysRevA.79.020305

    Article  ADS  Google Scholar 

  25. Burgarth, D., Ajoy, A.: Evolution-free Hamiltonian parameter estimation through zeeman markers. Phys. Rev. Lett. 119(3), 030402 (2017). https://doi.org/10.1103/PhysRevLett.119.030402

    Article  ADS  Google Scholar 

Download references

Acknowledgements

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

Author information

Authors and Affiliations

  1. Shanghai Advanced Research Institute, Chinese Academy of Sciences, Shanghai, 201210, China

    He Feng, Tian-Min Yan & Y. H. Jiang

  2. University of Chinese Academy of Sciences, Beijing, 100049, China

    He Feng & Y. H. Jiang

  3. ShanghaiTech University, Shanghai, 201210, China

    Y. H. Jiang

Authors
  1. He Feng
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Tian-Min Yan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Y. H. Jiang
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Tian-Min Yan.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work is supported by the National Natural Science Foundation of China (Grants Nos. 11420101003, 11604347, 11827806, 11874368 and 91636105). We also acknowledge the support from Shanghai-XFEL beamline project (SBP) and Shanghai high repetition rate XFEL and extreme light facility (SHINE).

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Feng, H., Yan, TM. & Jiang, Y.H. Dimerized decomposition of quantum evolution on an arbitrary graph. Quantum Inf Process 19, 30 (2020). https://doi.org/10.1007/s11128-019-2532-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-019-2532-y

Keywords

Search

Navigation