Perfect state transfer on weighted graphs of the Johnson scheme


We characterize quantum perfect state transfer on real-weighted graphs of the Johnson scheme \({\mathcal {J}}(n,k)\), which represent spin networks with non-nearest neighbor couplings. Given \({\mathcal {J}}(n,k)=\{A_{1},A_{2},\ldots ,A_{k}\}\) and \(A(X) = w_0A_0 + \cdots + w_m A_m\), we show that X has perfect state transfer at time \(\tau \) if and only if \(n=2k\), \(m\ge 2^{\lfloor {\log _2(k)} \rfloor }\), and there are integers \(c_{1},c_{2},\ldots ,c_{m}\) such that

  1. (i)

    \(c_j\) is odd if and only if j is a power of 2, and

  2. (ii)

    for \(r=1,2,\ldots ,m\),

    $$\begin{aligned} w_r = \frac{\pi }{\tau } \sum _{j=r}^m \frac{c_j}{\left( {\begin{array}{c}2j\\ j\end{array}}\right) } \left( {\begin{array}{c}k-r\\ j-r\end{array}}\right) . \end{aligned}$$

We then characterize perfect state transfer on unweighted graphs of \({\mathcal {J}}(n,k)\). In particular, we obtain a simple construction that generates all graphs of \({\mathcal {J}}(n,k)\) with perfect state transfer at time \(\pi /2\).

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  1. 1.

    Ahmadi, B., Shirdareh Haghighi, M.H., Mokhtar, A.: Perfect quantum state transfer on the Johnson scheme (2017). arXiv:1710.09096

  2. 2.

    Albanese, C., Christandl, M., Datta, N., Ekert, A.: Mirror inversion of quantum states in linear registers. Phys. Rev. Lett. 93(23), 230502 (2004)

    ADS  MathSciNet  Article  Google Scholar 

  3. 3.

    Bannai, E., Ito, T.: Algebraic Combinatorics. I. The Benjamin/Cummings Publishing Co. Inc, Menlo Park, CA (1984)

    MATH  Google Scholar 

  4. 4.

    Bhargava, M.: The factorial function and generalizations. Technical report 9 (2000)

  5. 5.

    Brouwer, A.E., Cohen, A.M., Neumaier, A.: Distance-Regular Graphs. Springer, Berlin (1989)

    Book  Google Scholar 

  6. 6.

    Brouwer, A.E., Haemers, W.H.: Spectra of Graphs. Universitext, Springer, New York (2012)

    Book  Google Scholar 

  7. 7.

    Chan, A.: Complex Hadamard matrices, instantaneous uniform mixing and cubes (2013). arXiv:1305.5811

  8. 8.

    Godsil, C.: Periodic graphs. Electron. J. Combin. 18(1), 23 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Christandl, M., Datta, N., Ekert, A., Landahl, A.J.: Perfect state transfer in quantum spin networks. Phys. Rev. Lett. 92(18), 187902 (2004)

    ADS  Article  Google Scholar 

  10. 10.

    Christandl, M., Vinet, L., Zhedanov, A.: Analytic next-to-nearest-neighbor X X models with perfect state transfer and fractional revival. Phys. Rev. A 96(3), 032335 (2017)

    ADS  Article  Google Scholar 

  11. 11.

    Coutinho, G., Godsil, C., Guo, K., Vanhove, F.: Perfect state transfer on distance-regular graphs and association schemes. Linear Algebra Appl. 478, 108–130 (2015)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Dickson, L.E.: History of the Theory of Numbers: Divisibility and Primality, vol. 1. Chelsea Publishing Co., New York (1966)

    MATH  Google Scholar 

  13. 13.

    Godsil, C.: When can perfect state transfer occur? Electron. J. Linear Algebra 23(1), 877–890 (2012)

    MathSciNet  MATH  Google Scholar 

  14. 14.

    Jafarizadeh, M.A., Sufiani, R.: Perfect state transfer over distance-regular spin networks. Phys. Rev. A 77(2), 022315 (2008)

    ADS  Article  Google Scholar 

  15. 15.

    Kay, A.: Perfect state transfer: beyond nearest-neighbor couplings. Phys. Rev. A 73(3), 032306 (2006)

    ADS  MathSciNet  Article  Google Scholar 

  16. 16.

    Kay, A.: Perfect, efficient, state transfer and its application as a constructive tool. Int. J. Quantum Inf. 08(04), 641–676 (2010)

    Article  Google Scholar 

  17. 17.

    Koekoek, R., Lesky, P.A., Swarttouw, R.F.: Hypergeometric Orthogonal Polynomials and Their q-Analogues. Springer Monographs in Mathematics. Springer, Berlin (2010)

    Book  Google Scholar 

  18. 18.

    Vinet, L., Zhedanov, A.: How to construct spin chains with perfect state transfer. Phys. Rev. A 85(1), 012323 (2012)

    ADS  Article  Google Scholar 

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We would like to thank the anonymous referees for their helpful comments.

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Correspondence to Hanmeng Zhan.

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Vinet, L., Zhan, H. Perfect state transfer on weighted graphs of the Johnson scheme. Lett Math Phys 110, 2491–2504 (2020).

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  • Quantum walks
  • Perfect state transfer
  • Beyond nearest neighbor couplings
  • Johnson scheme
  • Dual Hahn polynomial

Mathematics Subject Classification

  • 05C50
  • 05E30
  • 11A07
  • 11C20