Non-binary entanglement-assisted quantum stabilizer codes

非二元纠缠辅助量子稳定子码

Abstract

In this paper, we present the p m-ary entanglement-assisted (EA) stabilizer formalism, where p is a prime and m is a positive integer. Given an arbitrary non-abelian “stabilizer”, the problem of code construction and encoding is settled perfectly in the case of m = 1. The optimal number of required maximally entangled pairs is discussed and an algorithm to determine the encoding and decoding circuits is proposed. We also generalize several bounds on p-ary EA stabilizer codes, such as the BCH bound, the G-V bound and the linear programming bound. However, the issue becomes tricky when it comes to m > 1, in which case, the former construction method applies only when the non-commuting “stabilizer” satisfies a sophisticated limitation.

中文摘要

本文对pm元纠缠辅助量子稳定子码进行了深入研究, 其中p为素数, m为正整数。在m=1, 即p元码的情况下, 给定任意一个非交换的“稳定子”, 纠缠辅助码的构造及编码问题均得到了彻底解决。同时, 本文还计算了p元码构造时所需的最优最大纠缠对数目以及BCH界、G-V界、线性规划界等码界。此外, 本文还提出了确定p元纠缠辅助码编码及译码线路的确切算法。然而需要注意的是, 当m>1时, p元码的构造方法只有当所给的非交换的“稳定子”满足一个苛刻的条件时才能适用。

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References

  1. 1

    Calderbank A R, Shor P W. Good quantum error-correcting codes exist. Phys Rev A, 1995, 54: 1098–1105

    Article  Google Scholar 

  2. 2

    Gottesman D. Stabilizer codes and quantum error correction. Dissertation for Ph.D. Degree. Psadena: California Institute of Technology, 1997. 17–35

    Google Scholar 

  3. 3

    Steane A M. Error correcting codes in quantum theory. Phys Rev Lett, 1997, 77: 793–797

    MathSciNet  Article  MATH  Google Scholar 

  4. 4

    Gottesman D. An introduction to quantum error correction. In: Lomonaco S J, ed. Quantum Computation: a Grand Mathematical Challenge for the Twenty-First Century and the Millennium. Providence: American Mathematical Society, 2002. 221–235

    Google Scholar 

  5. 5

    Nielsen M A, Chuang I L. Quantum computation and quantum information. Am J Phys, 2002, 70: 558–559

    Article  Google Scholar 

  6. 6

    Young K C, Sarovar M, Blume-Kohout R, et al. Error suppression and error correction in adiabatic quantum computation: techniques and challenges. Phys Rev X, 2013, 3: 5326–5333

    Google Scholar 

  7. 7

    Lidar D, Brun T. Quantum Error Correction. Cambridge: Cambridge University Press, 2013. 181–199

    Google Scholar 

  8. 8

    Bowen G. Entanglement required in achieving entanglement-assisted channel capacities. Phys Rev A, 2002, 66: 357–364

    Article  Google Scholar 

  9. 9

    Brun T, Devetak I, Hsieh M H, et al. Catalytic quantum error correction. IEEE Trans Inf Theory, 2006, 60: 3073–3089

    MathSciNet  Article  Google Scholar 

  10. 10

    Brun T, Devetak I, Hsieh M H, et al. Correcting quantum errors with entanglement. Science, 2006, 314: 436–439

    MathSciNet  Article  MATH  Google Scholar 

  11. 11

    Wilde M M. Quantum coding with entanglement. Dissertation for Ph.D. Degree. Los Angeles: University of Southern California, 2008. 21–40

    Google Scholar 

  12. 12

    Lai C Y, Brun T. Entanglement increases the error-correcting ability of quantum error-correcting codes. Phys Rev A, 2013, 88: 2343–2347

    Google Scholar 

  13. 13

    Lai C Y, Brun T A, Wilde M M, et al. Duality in entanglement-assisted quantum error correction. IEEE Trans Inf Theory, 2013, 59: 4020–4024

    MathSciNet  Article  Google Scholar 

  14. 14

    Lai C Y, Brun T A, Wilde M M, et al. Dualities and identities for entanglement-assisted quantum codes. Quantum Inf Process, 2014, 13: 957–990

    MathSciNet  Article  MATH  Google Scholar 

  15. 15

    Bennett C H, Brassard G, Crepeau C, et al. Teleporting an unknown quantum state via dual classical and Einstein- Podolsky-Rosen channels. Phys Rev Lett, 1993, 70: 1895–1899

    MathSciNet  Article  MATH  Google Scholar 

  16. 16

    Bennett C H, Wiesner S J. Communication via one- and two-particle operators on Einstein-Podolsky-Rosen states. Phys Rev Lett, 1992, 69: 2881–2884

    MathSciNet  Article  MATH  Google Scholar 

  17. 17

    Blume-Kohout R, Caves C M, Deutsch I H, et al. Climbing mount scalable: physical-resource requirements for a scalable quantum computer. Found Phys, 2002, 32: 1641–1670

    MathSciNet  Article  Google Scholar 

  18. 18

    Soderberg K A B, Monroe C. Phonon-mediated entanglement for trapped ion quantum computing. Rep Prog Phys, 2010, 73: 569–580

    Article  Google Scholar 

  19. 19

    Bennett C H, Brassard G, Popescu S, et al. Purification of noisy entanglement and faithful teleportation via noisy channels. Phy Rev L, 1996, 76: 722–725

    Article  Google Scholar 

  20. 20

    Bennett C H, Divincenzo D P, Smolin J A, et al. Mixed-state entanglement and quantum error correction. Phy Rev A, 1996, 54: 3824–3851

    MathSciNet  Article  Google Scholar 

  21. 21

    Gottesman D. Fault-tolerant quantum computation with higher-dimensional systems. Chaos Soliton Fract, 1998, 10: 302–313

    MathSciNet  MATH  Google Scholar 

  22. 22

    Rains E M. Nonbinary quantum codes. IEEE Trans Inf Theory, 1999, 45: 1827–1832

    MathSciNet  Article  MATH  Google Scholar 

  23. 23

    Ashikhmin A, Knill E. Nonbinary quantum stabilizer codes. IEEE Trans Inf Theory, 2001, 47: 3065–3072

    MathSciNet  Article  MATH  Google Scholar 

  24. 24

    Grassl M, Roetteler M, Beth T, et al. Efficient quantum circuits for non-qubit quantum error-correcting codes. Int J Found Comput S, 2003, 14: 757–775

    MathSciNet  Article  MATH  Google Scholar 

  25. 25

    Grassl M, Beth T, Rotteler M, et al. On optimal quantum codes. Int J Quantum Inf, 2004, 2: 757–775

    Article  MATH  Google Scholar 

  26. 26

    Ketkar A, Klappenecker A, Kumar S, et al. Nonbinary stabilizer codes over finite fields. IEEE Trans Inf Theory, 2006, 52: 4892–4914

    MathSciNet  Article  MATH  Google Scholar 

  27. 27

    Kim J, Walker J. Nonbinary quantum error-correcting codes from algebraic curves. Discrete Math, 2008, 308: 3115–3124

    MathSciNet  Article  MATH  Google Scholar 

  28. 28

    Feng K Q, Chen H. Quantum Error-Correcting Codes. Beijing: Science Press, 2010. 103–106

    Google Scholar 

  29. 29

    Smith A, Anderson B E, Sosa-Martinez H, et al. Quantum control in the Cs 6S(1/2) ground manifold using radiofrequency and microwave magnetic fields. Phys Rev Lett, 2013, 111: 170502

    Article  Google Scholar 

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Correspondence to Zhi Ma.

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Luo, L., Ma, Z., Wei, Z. et al. Non-binary entanglement-assisted quantum stabilizer codes. Sci. China Inf. Sci. 60, 42501 (2017). https://doi.org/10.1007/s11432-015-0932-y

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Keywords

  • entanglement
  • non-commutativity
  • non-binary stabilizer codes
  • quantum error-correction
  • quantum circuits
  • bounds

关键词

  • 纠缠
  • 非对易
  • 非二元稳定子码
  • 量子纠错
  • 量子线路
  • 码界