Quantum Information Processing

, Volume 9, Issue 5, pp 591–610 | Cite as

Nonlocal quantum information in bipartite quantum error correction

Article

Abstract

We show how to convert an arbitrary stabilizer code into a bipartite quantum code. A bipartite quantum code is one that involves two senders and one receiver. The two senders exploit both nonlocal and local quantum resources to encode quantum information with local encoding circuits. They transmit their encoded quantum data to a single receiver who then decodes the transmitted quantum information. The nonlocal resources in a bipartite code are ebits and nonlocal information qubits, and the local resources are ancillas and local information qubits. The technique of bipartite quantum error correction is useful in both the quantum communication scenario described above and in fault-tolerant quantum computation. It has application in fault-tolerant quantum computation because we can prepare nonlocal resources offline and exploit local encoding circuits. In particular, we derive an encoding circuit for a bipartite version of the Steane code that is local and additionally requires only nearest-neighbor interactions. We have simulated this encoding in the CNOT extended rectangle with a publicly available fault-tolerant simulation software. The result is that there is an improvement in the “pseudothreshold” with respect to the baseline Steane code, under the assumption that quantum memory errors occur less frequently than quantum gate errors.

Keywords

Bipartite quantum error correction Entanglement-assisted quantum error correction Fault-tolerant quantum computation 

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. 1.
    Ahlswede R., Cai N.: A strong converse theorem for quantum multiple access channels. Electron. Notes Discret. Math. 21, 137–141 (2005) (General Theory of Information Transfer and Combinatorics)CrossRefMathSciNetGoogle Scholar
  2. 2.
    Aliferis P., Cross A.W.: Subsystem fault tolerance with the Bacon–Shor code. Phys. Rev. Lett. 98(22), 220502 (2007). doi:10.1103/PhysRevLett.98.220502 CrossRefADSPubMedGoogle Scholar
  3. 3.
    Aliferis P., Gottesman D., Preskill J.: Quantum accuracy threshold for concatenated distance-3 codes. Quantum Inf. Comput. 6, 97–165 (2006)MathSciNetMATHGoogle Scholar
  4. 4.
    Bacon D.: Operator quantum error correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73, 012,340 (2006)CrossRefGoogle Scholar
  5. 5.
    Bravyi S., Fattal D., Gottesman D.: GHZ extraction yield for multipartite stabilizer states. J. Math. Phys. 47, 062,106 (2006)CrossRefMathSciNetGoogle Scholar
  6. 6.
    Brun, T.A., Devetak, I., Hsieh, M.H.: Catalytic quantum error correction (2006). arXiv:quant-ph/0608027Google Scholar
  7. 7.
    Brun T.A., Devetak I., Hsieh M.H.: Correcting quantum errors with entanglement. Science 314(5798), 436–439 (2006)CrossRefMathSciNetADSPubMedGoogle Scholar
  8. 8.
    Calderbank A.R., Rains E.M., Shor P.W., Sloane N.J.A.: Quantum error correction via codes over GF(4). IEEE Trans. Inf. Theory 44, 1369–1387 (1998)CrossRefMathSciNetMATHGoogle Scholar
  9. 9.
    Cerf N.J., Cleve R.: Information-theoretic interpretation of quantum error-correcting codes. Phys. Rev. A 56(3), 1721–1732 (1997). doi:10.1103/PhysRevA.56.1721 CrossRefMathSciNetADSGoogle Scholar
  10. 10.
    Cirac J.I., Ekert A.K., Huelga S.F., Macchiavello C.: Distributed quantum computation over noisy channels. Phys. Rev. A 59(6), 4249–4254 (1999). doi:10.1103/PhysRevA.59.4249 CrossRefMathSciNetADSGoogle Scholar
  11. 11.
    Cover T.M., Thomas J.A.: Elements of Information Theory Series in Telecommunication. Wiley, New York (1991)CrossRefGoogle Scholar
  12. 12.
    Cross, A.W.: QASM fault-tolerant simulation software. http://www.media.mit.edu/quanta/quanta-web/projects/qasm-tools/ (2006)
  13. 13.
    Cross A.W., DiVincenzo D.P., Terhal B.M.: A comparative code study for quantum fault-tolerance. Quantum Inf. Comput. 9, 541–572 (2009)MathSciNetMATHGoogle Scholar
  14. 14.
    Devetak I.: The private classical capacity and quantum capacity of a quantum channel. IEEE Trans. Inf. Theory 51(1), 44–55 (2005)CrossRefMathSciNetGoogle Scholar
  15. 15.
    Devetak I., Harrow A.W., Winter A.: A resource framework for quantum Shannon theory. IEEE Trans. Inf. Theory 54(10), 4587–4618 (2008)CrossRefMathSciNetGoogle Scholar
  16. 16.
    Devetak I., Harrow A.W., Winter A.J.: A family of quantum protocols. Phys. Rev. Lett. 93, 239,503 (2004)MathSciNetGoogle Scholar
  17. 17.
    Devetak I., Shor P.W.: The capacity of a quantum channel for simultaneous transmission of classical and quantum information. Commun. Math. Phys. 256(2), 287–303 (2005)CrossRefMathSciNetADSMATHGoogle Scholar
  18. 18.
    Devetak I., Winter A.: Relating quantum privacy and quantum coherence: an operational approach. Phys. Rev. Lett. 93, 080,501 (2004)MathSciNetGoogle Scholar
  19. 19.
    Devetak I., Winter A.: Distillation of secret key and entanglement from quantum states. Proc. Roy. Soc. A 461, 207–235 (2005)CrossRefMathSciNetADSMATHGoogle Scholar
  20. 20.
    D’Hondt, E.: Distributed quantum computation: a measurement-based approach. Ph.D. thesis, Vrije Universiteit Brussel (2005)Google Scholar
  21. 21.
    Eisert J., Jacobs K., Papadopoulos P., Plenio M.B.: Optimal local implementation of nonlocal quantum gates. Phys. Rev. A 62(5), 052,317 (2000). doi:10.1103/PhysRevA.62.052317 CrossRefGoogle Scholar
  22. 22.
    Fattal, D., Cubitt, T.S., Yamamoto, Y., Bravyi, S., Chuang, I.L.: Entanglement in the stabilizer formalism (2004). Quant-ph/0406168Google Scholar
  23. 23.
    Gaitan F.: Quantum Error Correction and Fault Tolerant Quantum Computing. CRC Press, Taylor and Francis Group, Boca Raton (2008)MATHGoogle Scholar
  24. 24.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. Ph.D. thesis, California Institute of Technology (1997)Google Scholar
  25. 25.
    Grassl, M.: Encoding circuits for quantum error-correcting codes (Copyright 2007). http://iaks-www.ira.uka.de/home/grassl/QECC/circuits/index.html
  26. 26.
    Grassl, M., Rötteler, M.: Noncatastrophic encoders and encoder inverses for quantum convolutional codes. In: IEEE International Symposium on Information Theory (quant-ph/0602129), pp. 1109–1113. Seattle, Washington, USA (2006)Google Scholar
  27. 27.
    Grover, L.K.: Quantum Telecomputation. arXiv:quant-ph/9704012 (1997)Google Scholar
  28. 28.
    Harrow A.W.: Coherent communication of classical messages. Phys. Rev. Lett. 92, 097,902 (2004)MathSciNetGoogle Scholar
  29. 29.
    Horodecki M., Oppenheim J., Winter A.: Partial quantum information. Nature 436, 673–676 (2005)CrossRefADSPubMedGoogle Scholar
  30. 30.
    Horodecki M., Oppenheim J., Winter A.: Quantum state merging and negative information. Commun. Math. Phys. 269(1), 107–136 (2007)CrossRefMathSciNetADSMATHGoogle Scholar
  31. 31.
    Hsieh M.H., Devetak I., Winter A.: Entanglement-assisted capacity of quantum multiple-access channels. IEEE Trans. Inf. Theory 54(7), 3078–3090 (2008)CrossRefMathSciNetGoogle Scholar
  32. 32.
    Klimovitch, G.V., Winter, A.: Classical capacity of quantum binary adder channels. arXiv:quant-ph/0502055 (2005)Google Scholar
  33. 33.
    Kobayashi, H., Gall, F.L., Nishimura, H., Rötteler, M.: Perfect quantum network communication protocol based on classical network coding. arXiv:0902.1299 (2009)Google Scholar
  34. 34.
    Kremsky I., Hsieh M.H., Brun T.A.: Classical enhancement of quantum-error-correcting codes. Phys. Rev. A 78(1), 012341 (2008). doi:10.1103/PhysRevA.78.012341 CrossRefADSGoogle Scholar
  35. 35.
    Kribs D., Laflamme R., Poulin D.: A unified and generalized approach to quantum error correction. Phys. Rev. Lett. 94, 180,510 (2005)CrossRefGoogle Scholar
  36. 36.
    Leung, D., Oppenheim, J., Winter, A.: Quantum network communication—the butterfly and beyond. arXiv:quant-ph/0608223 (2006)Google Scholar
  37. 37.
    MacWilliams F.J., Sloane N.J.A.: The Theory of Error-Correcting Codes. Elsevier, Amsterdam (1977)MATHGoogle Scholar
  38. 38.
    Maslov D.: Linear depth stabilizer and quantum fourier transformation circuits with no auxiliary qubits in finite-neighbor quantum architectures. Phys. Rev. A 76(5), 052310 (2007). doi:10.1103/PhysRevA.76.052310 CrossRefADSGoogle Scholar
  39. 39.
    Meter III, R.D.V.: Architecture of a Quantum Multicomputer Optimized for Shor’s Factoring Algorithm. Ph.D. thesis, Keio University (2006). (quant-ph/0607065)Google Scholar
  40. 40.
    Nielsen M.A., Chuang I.L.: Quantum Computation and Quantum Information. Cambridge University Press, New York (2000)MATHGoogle Scholar
  41. 41.
    Poulin D.: Stabilizer formalism for operator quantum error correction. Phys. Rev. Lett. 95, 230,504 (2005)CrossRefGoogle Scholar
  42. 42.
    Shaw B., Wilde M.M., Oreshkov O., Kremsky I., Lidar D.: Encoding one logical qubit into six physical qubits. Phys. Rev. A 78, 012,337 (2008)CrossRefGoogle Scholar
  43. 43.
    Shor P.W.: Scheme for reducing decoherence in quantum computer memory. Phys. Rev. A 52, 2493–2496 (1995)CrossRefADSGoogle Scholar
  44. 44.
    Shor, P.W.: Fault tolerant quantum computation. In: Proceedings of the 37th Symposium on the Foundations of Computer Science, pp. 56–65. IEEE, Los Alamitos, CA (1996)Google Scholar
  45. 45.
    Steane A.M.: Active stabilization, quantum computation, and quantum state synthesis. Phys. Rev. Lett. 78(11), 2252–2255 (1997). doi:10.1103/PhysRevLett.78.2252 CrossRefADSGoogle Scholar
  46. 46.
    Svore K.M., Cross A.W., Chuang I.L., Aho A.V.: A flow-map model for analyzing pseudothresholds in fault-tolerant quantum computing. Quantum Inf. Comput. 6(3), 193–212 (2006)MathSciNetMATHGoogle Scholar
  47. 47.
    Viterbi A.J.: CDMA: Principles of Spread Spectrum Communication. 1st edn. Prentice Hall PTR, Upper Saddle River (1995)MATHGoogle Scholar
  48. 48.
    Wilde M.M.: Logical operators of quantum codes. Phys. Rev. A 79, 062,322 (2009)MathSciNetGoogle Scholar
  49. 49.
    Wilde, M.M., Brun, T.A.: Entanglement-assisted quantum convolutional coding (2007). arXiv:0712.2223Google Scholar
  50. 50.
    Wilde M.M., Brun T.A.: Optimal entanglement formulas for entanglement-assisted quantum coding. Phys. Rev. A 77, 064,302 (2008)Google Scholar
  51. 51.
    Wilde, M.M., Brun, T.A.: Quantum convolutional coding with shared entanglement: General structure (2008). arXiv:0807.3803Google Scholar
  52. 52.
    Wilde, M.M., Brun, T.A.: Unified quantum convolutional coding. In: Proceedings of the IEEE International Symposium on Information Theory, pp. 359–363 (2008). (arXiv:0801.0821)Google Scholar
  53. 53.
    Wilde M.M., Brun T.A., Dowling J.P., Lee H.: Coherent communication with linear optics. Phys. Rev. A 77(2), 022321 (2008). doi:10.1103/PhysRevA.77.022321 CrossRefADSGoogle Scholar
  54. 54.
    Wilde M.M., Krovi H., Brun T.A.: Coherent communication with continuous quantum variables. Phys. Rev. A 75(6), 060303 (2007). doi:10.1103/PhysRevA.75.060303 CrossRefMathSciNetADSGoogle Scholar
  55. 55.
    Wilde, M.M., Krovi, H., Brun, T.A.: Convolutional entanglement distillation (2007). arXiv:0708.3699Google Scholar
  56. 56.
    Winter A.: The capacity of the quantum multiple access channel. IEEE Trans. Inf. Theory 47, 3059–3065 (2001)CrossRefMathSciNetMATHGoogle Scholar
  57. 57.
    Yard, J.: Simultaneous Classical-Quantum Capacities of Quantum Multiple Access Channels. Ph.D. thesis, Stanford University, Stanford, CA (2005). Quant-ph/0506050Google Scholar
  58. 58.
    Yard, J., Hayden, P., Devetak, I.: Quantum broadcast channels (2006). arXiv:quant-ph/0603098Google Scholar
  59. 59.
    Yard J., Hayden P., Devetak I.: Capacity theorems for quantum multiple-access channels: Classical-quantum and quantum-quantum capacity regions. IEEE Trans. Inf. Theory 54(7), 3091–3113 (2008)CrossRefMathSciNetGoogle Scholar
  60. 60.
    Yeung R.W.: A First Course in Information Theory. Springer, New York (2002)Google Scholar

Copyright information

© Springer Science+Business Media, LLC 2010

Authors and Affiliations

  1. 1.School of Computer ScienceMcGill UniversityMontrealCanada
  2. 2.Information and Quantum Systems LaboratoryHewlett-Packard LaboratoriesPalo AltoUSA

Personalised recommendations