Quantum Information Processing

, Volume 14, Issue 8, pp 2775–2818 | Cite as

Dual-code quantum computation model



In this work, we propose the dual-code quantum computation model—a fault-tolerant quantum computation scheme which alternates between two different quantum error-correction codes. Since the chosen two codes have different sets of transversal gates, we can implement a universal set of gates transversally, thereby reducing the overall cost. We use code teleportation to convert between quantum states in different codes. The overall cost is decreased if code teleportation requires fewer resources than the fault-tolerant implementation of the non-transversal gate in a specific code. To analyze the cost reduction, we investigate two cases with different base codes, namely the Steane and Bacon-Shor codes. For the Steane code, neither the proposed dual-code model nor another variation of it achieves any cost reduction since the conventional approach is simple. For the Bacon-Shor code, the three proposed variations of the dual-code model reduce the overall cost. However, as the encoding level increases, the cost reduction decreases and becomes negative. Therefore, the proposed dual-code model is advantageous only when the encoding level is low and the cost of the non-transversal gate is relatively high.


Fault-tolerant quantum computation Steane code Bacon-Shor code Reed-Muller code Code conversion Teleportation 


  1. 1.
    Aharonov, D., Ben-Or, M.: Fault-tolerant quantum computation with constant error rate. SIAM J. Comput. 38(4), 1207–1282 (2008)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Aliferis, P., Gottesman, D., Preskill, J.: Quantum accuracy threshold for concatenated distance-3 codes. Quant. Inf. Comput. 6, 97Google Scholar
  3. 3.
    Anderson, J.T., Duclos-Cianci, G., Poulin, D.: Fault-tolerant conversion between the steane and reed-muller quantum codes. Phys. Rev. Lett. 113, 080501 (2014)ADSCrossRefGoogle Scholar
  4. 4.
    Bacon, D.: Operator quantum error-correcting subsystems for self-correcting quantum memories. Phys. Rev. A 73(1), 012340 (2006)MathSciNetADSCrossRefGoogle Scholar
  5. 5.
    Bombin, H.: Clifford gates by code deformation. N. J. Phys. 13(4), 043005 (2011)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Bombin, H.: Optimal transversal gates under geometric constraints. ArXiv e-prints, Nov (2013)Google Scholar
  7. 7.
    Bombin, H., Martin-Delgado, M.A.: Quantum measurements and gates by code deformation. J. Phys. A Math. Theor. 42(9), 095302 (2009)MathSciNetADSCrossRefGoogle Scholar
  8. 8.
    Bravyi, S., Haah, J.: Magic-state distillation with low overhead. Phys. Rev. A 86, 052329 (2012)ADSCrossRefGoogle Scholar
  9. 9.
    Bravyi, S., Kitaev, A.: Universal quantum computation with ideal clifford gates and noisy ancillas. Phys. Rev. A 71(2), 022316 (2005)MathSciNetADSCrossRefGoogle Scholar
  10. 10.
    Brun, T.: Fault-tolerant, nondestructive measurement of logical operators and quantum teleportation in large stabilizer codes. In: APS Meeting Abstracts, March, p. 27007 (2013)Google Scholar
  11. 11.
    Buchbinder, S.D., Huang, C.L., Weinstein, Y.S.: Encoding an arbitrary state in a [7,1,3] quantum error correction code. Quantum Inf. Process. 12, 699–719 (2013)MathSciNetADSCrossRefGoogle Scholar
  12. 12.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54(2), 1098–1105 (1996)ADSCrossRefGoogle Scholar
  13. 13.
    Campbell, E.T., Browne, D.E.: Bound states for magic state distillation in fault-tolerant quantum computation. Phys. Rev. Lett. 104(3), 030503 (2010)ADSCrossRefGoogle Scholar
  14. 14.
    Choi, B.-S.: Cost comparison between code teleportation and stabilizer sequence methods for code conversion. In: Proceedings of the International Conference on ICT Convergence (ICTC 2013), Special Session for Quantum Information Processing, pp. 1083–1087, October 2013Google Scholar
  15. 15.
    Cleve, R., Gottesman, D.: Efficient computations of encodings for quantum error correction. Phys. Rev. A 56, 76–82 (1997)ADSCrossRefGoogle Scholar
  16. 16.
    Dennis, E., Kitaev, A., Landahl, A., Preskill, J.: Topological quantum memory. J. Math. Phys. 43(9), 4452–4505 (2002)MathSciNetADSCrossRefGoogle Scholar
  17. 17.
    Dumer, I., Kovalev, A.A., Pryadko, L.P.: Thresholds for correcting errors, erasures, and faulty syndrome measurements in degenerate quantum codes. ArXiv e-prints (2014)Google Scholar
  18. 18.
    Eastin, B., Knill, E.: Restrictions on transversal encoded quantum gate sets. Phys. Rev. Lett. 102(11), 110502 (2009)ADSCrossRefGoogle Scholar
  19. 19.
    Fowler, A.G., Devitt, S.J., Jones, C.: Surface code implementation of block code state distillation. Sci. Rep. 3 (2013). doi: 10.1038/srep01939
  20. 20.
    Fowler, A.G., Stephens, A.M., Groszkowski, P.: High-threshold universal quantum computation on the surface code. Phys. Rev. A 80(5), 052312 (2009)ADSCrossRefGoogle Scholar
  21. 21.
    Gottesman, D.: The Heisenberg representation of quantum computers. arXiv:quant-ph/9807006 (1998)
  22. 22.
    Gottesman, D.: Fault-tolerant quantum computation with constant overhead. ArXiv e-prints, October (2013)Google Scholar
  23. 23.
    Gottesman, D.: Theory of fault-tolerant quantum computation. Phys. Rev. A 57(1), 127–137 (1998)MathSciNetADSCrossRefGoogle Scholar
  24. 24.
    Gottesman, D., Chuang, I.L.: Demonstrating the viability of universal quantum computation using teleportation and single-qubit operations. Nature 402(6760), 390–393 (1999). 1125ADSCrossRefGoogle Scholar
  25. 25.
    Jochym-O’Connor, T., Yu, Y., Helou, B., Laflamme, R.: The robustness of magic state distillation against errors in Clifford Gates. ArXiv e-prints, May 2012Google Scholar
  26. 26.
    Jochym-O’Connor, T., Laflamme, R.: Using concatenated quantum codes for universal fault-tolerant quantum gates. Phys. Rev. Lett. 112, 010505 (2014)ADSCrossRefGoogle Scholar
  27. 27.
    Jones, C.: Multilevel Distillation of magic states for quantum computing. ArXiv e-prints (2012)Google Scholar
  28. 28.
    Jones, C.: Low-overhead constructions for the fault-tolerant Toffoli gate. Phys. Rev. A 87(2), 022328 (2013)ADSCrossRefGoogle Scholar
  29. 29.
    Julia, K., Oded, R., Falk, U., Wolf, R.: Upper bounds on the noise threshold for fault-tolerant quantum computing. In: Proceedings of the 35th International Colloquium on Automata, Languages and Programming, Part I, ICALP ’08, pp. 845–856 (2008)Google Scholar
  30. 30.
    Knill, E.: Fault-tolerant postselected quantum computation: schemes. arXiv:quant-ph/0402171 (2004)
  31. 31.
    Knill, E.: Fault-tolerant postselected quantum computation: threshold analysis. arXiv:quant-ph/0404104 (2004)
  32. 32.
    Knill, E., Laflamme, R., Zurek, W.: Threshold accuracy for quantum computation. arXiv:quant-ph/9610011 (1996)
  33. 33.
    Knill, E., Laflamme, R.: Concatenated quantum codes. arXiv:quant-ph/9608012 (1996)
  34. 34.
    Knill, E., Laflamme, R., Wojciech H.Z.: Resilient quantum computation: error models and thresholds. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 454(1969):365–384 (1998)Google Scholar
  35. 35.
    Metodi, T.S., Faruque, A.I., Chong, F.T.: Quantum Computing for Computer Architects, 2nd Edition. Synthesis Lectures on Computer Architecture Series. Morgan & Claypool, USA (2011)Google Scholar
  36. 36.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information, 1st edn. Cambridge University Press, Cambridge (2000)Google Scholar
  37. 37.
    Oskin, M., Chong, F.T., Chuang, I.L.: A practical architecture for reliable quantum computers. IEEE Comput. 35(1), 79–87 (2002)CrossRefGoogle Scholar
  38. 38.
    Paetznick, A., Reichardt, B.W.: Universal fault-tolerant quantum computation with only transversal gates and error correction. Phys. Rev. Lett. 111, 090505 (2013)ADSCrossRefGoogle Scholar
  39. 39.
    Preskill, J.: Reliable quantum computers. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 454(1969), 385–410 (1998)Google Scholar
  40. 40.
    Reichardt, B.W.: Quantum universality by state distillation. Quantum Inf. Comput. 9(11), 1030–1052 (2011)MathSciNetGoogle Scholar
  41. 41.
    Shor, P. W.: Fault-tolerant quantum computation. In: Proceedings of the 37th Annual Symposium on Foundations of Computer Science, FOCS ’96, pp. 56, IEEE Computer Society, Washington, DC (1996)Google Scholar
  42. 42.
    Shor, P.W.: Fault-tolerant quantum computation. In: Annual Symposium on Foundations of Computer, Science, pp. 56–65, October, 1996Google Scholar
  43. 43.
    Steane, A.: Quantum Reed-Muller codes. arXiv:quant-ph/9608026 (1996)
  44. 44.
    Steane, A.: Multiple-particle interference and quantum error correction. Proc. R. Soc. Lond. Ser. A: Math. Phys. Eng. Sci. 452(1954), 2551–2577 (1996)Google Scholar
  45. 45.
    Steane, A.M.: Overhead and noise threshold of fault-tolerant quantum error correction. Phys. Rev. A 68(4), 042322 (2003)ADSCrossRefGoogle Scholar
  46. 46.
    Stephens, A.M.: Efficient fault-tolerant decoding of topological color codes. ArXiv e-prints, Feb 2014Google Scholar
  47. 47.
    Thaker, D.D., Metodi, T.S., Cross, A.W., Chuang, I.L., Chong, F.T.: Quantum memory hierarchies: efficient designs to match available parallelism in quantum computing. In: The International Symposium on Computer Architecture (ISCA), 2006, pp. 378–390 (2006)Google Scholar
  48. 48.
    Unger, F.: Noise threshold for universality of two-input gates. IEEE Trans. Inf. Theory 54(8), 3693–3698 (2008)MathSciNetADSCrossRefGoogle Scholar
  49. 49.
    Van den Nest, M.: Classical simulation of quantum computation, the Gottesman–Knill theorem, and slightly beyond. Quantum Inf. Comput. 10(3–4), 0258–0271 (2010)Google Scholar
  50. 50.
    Wang, D.S., Fowler, A.G., Hill, C.D., Hollenberg, L.C.L.: Graphical algorithms and threshold error rates for the 2D color code. Quantum Inf. Comput. 10(9), 780–802 (2010)Google Scholar
  51. 51.
    Wang, D.S., Fowler, A.G., Hollenberg, L.C.L.: Surface code quantum computing with error rates over 1. Phys. Rev. A 83, 020302 (2011)ADSCrossRefGoogle Scholar
  52. 52.
    Weinstein, Y.S.: Fidelity of an encoded [7,1,3] logical zero. Phys. Rev. A 84, 012323 (2011)ADSCrossRefGoogle Scholar
  53. 53.
    Zeng, B., Cross, A., Chuang, I.L.: Transversality versus universality for additive quantum codes. IEEE Trans. Inf. Theory 57(9), 6272–6284 (2011)MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 2015

Authors and Affiliations

  1. 1.Department of Applied PhysicsUniversity of TokyoTokyoJapan

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