Perfect state transfer in Laplacian quantum walk


For a graph G and a related symmetric matrix M, the continuous-time quantum walk on G relative to M is defined as the unitary matrix \(U(t) = \exp (-itM)\), where t varies over the reals. Perfect state transfer occurs between vertices u and v at time \(\tau \) if the (uv)-entry of \(U(\tau )\) has unit magnitude. This paper studies quantum walks relative to graph Laplacians. Some main observations include the following closure properties for perfect state transfer. If an n-vertex graph has perfect state transfer at time \(\tau \) relative to the Laplacian, then so does its complement if \(n\tau \in 2\pi {\mathbb {Z}}\). As a corollary, the join of \(\overline{K}_{2}\) with any m-vertex graph has perfect state transfer relative to the Laplacian if and only if \(m \equiv 2\pmod {4}\). This was previously known for the join of \(\overline{K}_{2}\) with a clique (Bose et al. in Int J Quant Inf 7:713–723, 2009). If a graph G has perfect state transfer at time \(\tau \) relative to the normalized Laplacian, then so does the weak product \(G \times H\) if for any normalized Laplacian eigenvalues \(\lambda \) of G and \(\mu \) of H, we have \(\mu (\lambda -1)\tau \in 2\pi {\mathbb {Z}}\). As a corollary, a weak product of \(P_{3}\) with an even clique or an odd cube has perfect state transfer relative to the normalized Laplacian. It was known earlier that a weak product of a circulant with odd integer eigenvalues and an even cube or a Cartesian power of \(P_{3}\) has perfect state transfer relative to the adjacency matrix. As for negative results, no path with four vertices or more has antipodal perfect state transfer relative to the normalized Laplacian. This almost matches the state of affairs under the adjacency matrix (Godsil in Discret Math 312(1):129–147, 2011).

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  1. 1.

    This is apparently a common technique in statistical and quantum physics. See [26, 27] for an application of this method to continuous-time random and quantum walks.

  2. 2.

    We follow a convention used by Mike Newman [30].

  3. 3.

    Doob [14] showed that \(-2 \in {{\mathrm{Spec}}}(\ell (G))\) if and only if G contains an even cycle or two odd cycles in the same component.


  1. 1.

    Aldous, D., Fill, J.: Reversible Markov Chains and Random Walks on Graphs. Monograph (2002)

  2. 2.

    Angeles-Canul, R.J., Norton, R., Opperman, M., Paribello, C., Russell, M., Tamon, C.: Perfect state transfer, integral circulants and join of graphs. Quantum Inf. Comput. 10(3&4), 325–342 (2010)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Bachman, R., Fredette, E., Fuller, J., Landry, M., Opperman, M., Tamon, C., Tollefson, A.: Perfect state transfer on quotient graphs. Quantum Inf. Comput. 10(3&4), 293–313 (2012)

    MathSciNet  MATH  Google Scholar 

  4. 4.

    Bayat, A., Banchi, L., Bose, S., Verruchi, P.: Initializing an unmodulated spin chain to operate as a high-quality data bus. Phys. Rev. A 83, 062328 (2011)

    Article  Google Scholar 

  5. 5.

    Bose, S.: Quantum communication through an unmodulated spin chain. Phys. Rev. Lett. 91(20), 207901 (2003)

    Article  Google Scholar 

  6. 6.

    Bose, S., Casaccino, A., Mancini, S., Severini, S.: Communication in XYZ all-to-all quantum networks with a missing link. Int. J. Quantum Inf. 7(4), 713–723 (2009)

    Article  MATH  Google Scholar 

  7. 7.

    Cardoso, D., Delorme, C., Rama, P.: Laplacian eigenvectors and eigenvalues and almost equitable partitions. Eur. J. Comb. 28(3), 665–673 (2007)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Childs, A.: Universal computation by quantum walk. Phys. Rev. Lett. 102, 180501 (2009)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Childs, A., Cleve, R., Deotto, E., Farhi, E., Gutmann, S., Spielman, D.: Exponential algorithmic speedup by a quantum walk. In: Proceedings of the 35th ACM Symposium on Theory of Computing, pp. 59–68 (2003)

  10. 10.

    Christandl, M., Datta, N., Dorlas, T., Ekert, A., Kay, A., Landahl, A.: Perfect transfer of arbitrary states in quantum spin networks. Phys. Rev. A 71, 032312 (2005)

    Article  Google Scholar 

  11. 11.

    Christandl, M., Datta, N., Ekert, A., Landahl, A.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92, 187902 (2004)

    Article  Google Scholar 

  12. 12.

    Chung, F.R.K.: Spectral Graph Theory. American Mathematical Society, Providence (1996)

    Google Scholar 

  13. 13.

    Coutinho, G., Liu, H.: No Laplacian perfect state transfer in trees. arXiv:1408.2935 [math.CO]

  14. 14.

    Doob, M.: An interrelation between line graphs, eigenvalues, and matroids. J. Comb. Theory Ser. B 15, 40–50 (1973)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Farhi, E., Goldstone, J., Gutmann, S.: A quantum algorithm for the hamiltonian NAND tree. Theory Comput. 4(8), 169–190 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Farhi, E., Gutmann, S.: Quantum computation and decision trees. Phys. Rev. A 58, 915–928 (1998)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Ge, Y., Greenberg, B., Perez, O., Tamon, C.: Perfect state transfer, graph products and equitable partitions. Int. J. Quantum Inf. 9(3), 823–842 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Godsil, C.: Graph Spectra and Quantum Walks. Manuscript (2014)

  19. 19.

    Godsil, C.: State transfer on graphs. Discret. Math. 312(1), 129–147 (2011)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Godsil, C.: Controllable subsets in graphs. Ann. Comb. 16, 733–744 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  21. 21.

    Godsil, C.: When can perfect state transfer occur? Electron. J. Linear Algebra 23, 877–890 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  22. 22.

    Godsil, C., Royle, G.: Algebraic Graph Theory. Springer, Berlin (2001)

    Google Scholar 

  23. 23.

    Godsil, C., Severini, S.: Control by quantum dynamics on graphs. Phys. Rev. A 81, 052316 (2010)

    MathSciNet  Article  Google Scholar 

  24. 24.

    Graham, R., Knuth, D., Patashnik, O.: Concrete Mathematics, 2nd edn. Addison-Wesley, Reading (1994)

    Google Scholar 

  25. 25.

    Grimmett, G., Stirzaker, D.: Probability and Random Processes. Oxford University Press, Oxford (1982)

    Google Scholar 

  26. 26.

    Grünbaum, A., Vinet, L., Zhedanov, A.: Birth and death processes and quantum spin chains. J. Math. Phys. 54, 062101 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  27. 27.

    Ide, Y., Konno, N.: Continuous-time quantum walks on the threshold network model. Math. Struct. Comput. Sci. 20(6), 1079–1090 (2010)

    MathSciNet  Article  MATH  Google Scholar 

  28. 28.

    Lin, Y., Lippner, G., Yau, S.T.: Quantum tunneling on graphs. Commun. Math. Phys. 311, 113–132 (2012)

    MathSciNet  Article  MATH  Google Scholar 

  29. 29.

    Moore, C., Russell, A.: Quantum walks on the hypercube. In: Proceedings of the 6th International Workshop on Randomization and Approximation in Computer Science, volume 2483 of Lecturer Notes in Computer Science, pp. 164–178. Springer, Berlin (2002)

  30. 30.

    Newman, M.W.: The Laplacian Spectrum of Graphs. Master’s thesis, University of Manitoba, Winnipeg, Canada (2000)

  31. 31.

    Niven, I.: Irrational Numbers. The Mathematical Association of America, Washington, DC (1965)

    Google Scholar 

  32. 32.

    Spielman, D.: Spectral graph theory. In: Naumann, U., Schenk, O. (eds.) Combinatorial Scientific Computing. Chapman and Hall/CRC, Boca Raton (2012)

    Google Scholar 

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We would like to thank Ada Chan, Gabriel Coutinho and Chris Godsil for their generous and helpful comments. We also thank the anonymous reviewers for comments and suggestions which improve this paper, and for pointing out the work of Bayat et al. [4]. The research of R.A., S.D., B.L., J.M. and C.T. was supported by a National Science Foundation Grant DMS-1262737 and a National Security Agency Grant H98230-14-1-0141. The research of H.Z. is supported by a Graduate Student Fellowship at the University of Waterloo while working the guidance of Chris Godsil.

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Correspondence to Christino Tamon.

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Alvir, R., Dever, S., Lovitz, B. et al. Perfect state transfer in Laplacian quantum walk. J Algebr Comb 43, 801–826 (2016).

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  • Laplacian
  • Quantum walk
  • Perfect state transfer
  • Join
  • Equitable partition
  • Weak product

Mathematics Subject Classification

  • MC05C50