Quantum network coding for multi-unicast problem based on 2D and 3D cluster states


We mainly consider quantum multi-unicast problem over directed acyclic network, where each source wishes to transmit an independent message to its target via bottleneck channel. Taking the advantage of global entanglement state 2D and 3D cluster states, these problems can be solved efficiently. At first, a universal scheme for the generation of resource states among distant communication nodes is provided. The corresponding between cluster and bigraph leads to a constant temporal resource cost. Furthermore, a new approach based on stabilizer formalism to analyze the solvability of several underlying quantum multi-unicast networks is presented. It is found that the solvability closely depends on the choice of stabilizer generators for a given cluster state. And then, with the designed measurement basis and parallel measurement on intermediate nodes, we propose optimal protocols for these multi-unicast sessions. Also, the analysis reveals that the resource consumption involving spatial resources, operational resources and temporal resources mostly reach the lower bounds.

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  1. 1

    Ahlswede R, Cai N, Li S Y R, et al. Network information flow. IEEE Trans Inf Theory, 2000, 46: 1204–1216

    MathSciNet  Article  MATH  Google Scholar 

  2. 2

    Shen J, Tan H, Wang J, et al. A novel routing protocol providing good transmission reliability in underwater sensor networks. J Internet Tech, 2015, 16: 171–178

    Google Scholar 

  3. 3

    Li S Y R, Yeung R W, Cai N. Linear network coding. IEEE Trans Inf Theory, 2003, 49: 371–381

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Xie S, Wang Y. Construction of tree network with limited delivery latency in homogeneous wireless sensor networks. Wirel Pers Commun, 2014, 78: 231–246

    Article  Google Scholar 

  5. 5

    Xiahou T, Li Z, Wu C. Information multicast in (pseudo-) planar networks: efficient network coding over small finite fields. In: Proceedings of the 2013 IEEE International Symposium on Network Coding, Galary, 2013. 1–6

    Google Scholar 

  6. 6

    Ding L H, Wu P, Wang H, et al. Lifetime maximization routing with network coding in wireless multihop networks. Sci China Inf Sci, 2013, 56: 022303

    MathSciNet  Google Scholar 

  7. 7

    Zhang C S, Ge J H, Li J, et al. Robust power allocation algorithm for analog network coding with imperfect CSI. Sci China Inf Sci, 2014, 57: 042312

    Google Scholar 

  8. 8

    Pan Z, Zhang Y, Kwong S. Efficient motion and disparity estimation optimization for low complexity multiview video coding. IEEE Trans Broadcast, 2015, 61: 166–176

    Article  Google Scholar 

  9. 9

    Wootters W K, Zurek W H. A single quantum cannot be cloned. Nature, 1982, 299: 802–803

    Article  Google Scholar 

  10. 10

    Hayashi M, Iwama K, Nishimura H, et al. Quantum network coding. In: Proceedings of the 24th Annual Conference on Theoretical Aspects of Computer Science. Berlin: Springer-Verlag, 2007. 610–621

    Google Scholar 

  11. 11

    Leung D, Oppenheim J, Winter A. Quantum network communication–the butterfly and beyond. IEEE Trans Inf Theory, 2010, 56: 3478–3490

    MathSciNet  Article  Google Scholar 

  12. 12

    Hayashi M. Prior entanglement between senders enables perfect quantum network coding with modification. Phys Rev A, 2007, 76: 040301

    MathSciNet  Article  Google Scholar 

  13. 13

    Luo M X, Xu G, Chen X B, et al. Efficient quantum transmission in multiple-source networks. Sci Rep, 2014, 4: 4571

    Google Scholar 

  14. 14

    Nishimura H. Quantum network coding and the current status of its studies. In: Proceedings of IEEE International Symposium on Information Theory and its Applications, Melbourne, 2014. 331–334

    Google Scholar 

  15. 15

    Satoh T, Le Gall F, Imai H. Quantum network coding for quantum repeaters. Phys Rev A, 2012, 86: 032331

    Article  Google Scholar 

  16. 16

    Ma S Y, Chen X B, Luo M X, et al. Probabilistic quantum network coding of M-qudit states over the butterfly network. Opt Commun, 2010, 283: 497–501

    Article  Google Scholar 

  17. 17

    Soeda A, Kinjo Y, Turner P S, et al. Quantum computation over the butterfly network. Phys Rev A, 2011, 84: 012333

    Article  Google Scholar 

  18. 18

    Akibue S, Murao M. Network coding for distributed quantum computation over cluster and butterfly networks. ArXiv: 1503.07740, 2015

    Google Scholar 

  19. 19

    Kobayashi H, Le Gall F, Nishimura H, et al. General scheme for perfect quantum network coding with free classical communication. In: Proceedings of the 36th International Colloquium on Automata, Languages and Programming, Lecture Note in Computer Science, Greece, 2009. 622–633

    Google Scholar 

  20. 20

    Kobayashi H, Le Gall F, Nishimura H, et al. Perfect quantum network communication protocol based on classical network coding. In: Proceedings of the 2010 IEEE International Symposium on Information Theory, Austin, 2010. 2686–2690

    Google Scholar 

  21. 21

    Kobayashi H, Le Gall F, Nishimura H, et al. Constructing quantum network coding schemes from classical nonlinear protocols. In: Proceedings of the 2011 IEEE International Symposium on Information Theory, Saint-Petersburg, 2011. 109–113

    Google Scholar 

  22. 22

    Li J, Chen X B, Xu G, et al. Perfect quantum network coding independent of classical network solutions. IEEE Commun Lett, 2015, 19: 115–118

    Article  Google Scholar 

  23. 23

    de Beaudrap N, Roetteler M. Quantum linear network coding as one-way quantum computation. In: Proceedings of the 9th Conference on the Theory of Quantum Computation, Communication and Cryptography, Singapore, 2014. 217–233

    Google Scholar 

  24. 24

    Raussendorf R, Browne D E, Briegel H J. Measurement-based quantum computation on cluster states. Phys Rev A, 2003, 68: 022312

  25. 25

    Briegel H J, Browne D E, Dür W, et al. Measurement-based quantum computation. Nature Phys, 2009, 5: 19–26

    Article  Google Scholar 

  26. 26

    Zwerger M, Briegel H J, Dür W. Measurement-based quantum communication. ArXiv: 1506.00985, 2015

    Google Scholar 

  27. 27

    Epping M, Kampermann H, Bruß D. Graph state quantum repeater networks. ArXiv: 1504.06599, 2015

    Google Scholar 

  28. 28

    Briegel H J, Raussendorf R. Persistent entanglement in arrays of interacting particles. Phys Rev Lett, 2001, 86: 910–913

    Article  Google Scholar 

  29. 29

    Nielsen M A, Chuang I L. Quantum Computation and Quantum Information. 10th ed. New York: Cambridge University Press, 2010

    Google Scholar 

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Correspondence to Xingming Sun.

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Li, J., Chen, X., Sun, X. et al. Quantum network coding for multi-unicast problem based on 2D and 3D cluster states. Sci. China Inf. Sci. 59, 042301 (2016). https://doi.org/10.1007/s11432-016-5539-3

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  • quantum information
  • network coding
  • multi-unicast
  • cluster state
  • stabilizer formalism