Skip to main content
SpringerLink
Log in

Euclidean and hyperbolic asymmetric topological quantum codes

  • Published:
Quantum Information Processing Aims and scope Submit manuscript

Abstract

In the last three decades, several constructions of quantum error-correcting codes were presented in the literature. Among these codes, there are the asymmetric ones, i.e., quantum codes whose Z-distance \(d_z\) is different from its X-distance \(d_x\). The topological quantum codes form an important class of quantum codes, where the toric code, introduced by Kitaev, was the first family of this type. After Kitaev’s toric code, several authors focused attention on investigating its structure and the constructions of new families of topological quantum codes over Euclidean and hyperbolic surfaces. As a consequence of establishing the existence and the construction of asymmetric topological quantum codes in Theorem 5.1, the main result of this paper, we introduce the class of hyperbolic asymmetric codes. Hence, families of Euclidean and hyperbolic asymmetric topological quantum codes are presented. An analysis regarding the asymptotic behavior of their distances \(d_x\) and \(d_z\) and encoding rates k/n versus the compact orientable surface’s genus is provided due to the significant difference between the asymmetric distances \(d_x\) and \(d_z\) when compared with the corresponding parameters of topological codes generated by other tessellations. This inherent unequal error protection is associated with the nontrivial homological cycle of the \(\{r,s\}\) tessellation and its dual, which may be appropriately explored depending on the application, where \(r\ne s\) and \((r-2)(s-2)\ge 4\). Three families of codes derived from the \(\{7,3\}\), \(\{5,4\}\), and \(\{10,5\}\) tessellations are highlighted.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9
Fig. 10

Similar content being viewed by others

Data Availability

Data sharing not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. Note that in all figures where the genus is considered as an axis of the graphic, we are writing the graphic of the function continuously only to illustrative purposes, but, in fact, the genus is discrete.

References

  1. Albuquerque, C.D., Palazzo, R., Jr., Silva, E.B.: Topological quantum codes on compact surfaces with genus \(g \ge 2\). J. Math. Phys. 50 023513-1–20 (2009)

  2. Bombin, H., Martin-Delgado, M.A.: Topological quantum distillation. Phys. Rev. Lett. 97, 180501 (2006)

    Article  ADS  Google Scholar 

  3. Bombin, H., Martin-Delgado, M.A.: Homological error correction: classical and quantum codes. J. Math. Phys. 48, 052105 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  4. Breuckmann, N.P., Terhal, B.M.: Constructions and noise threshold of hyperbolic surface codes. IEEE Trans. Inform. Theory 62(6), 3731–3744 (2016)

    Article  MathSciNet  Google Scholar 

  5. Breuckmann, N.P., Vuillot, C., Campbell, E., Krishna, A., Terhal, B.M.: Hyperbolic and semi-hyperbolic surface codes for quantum storage. e-print arXiv:1703.00590

  6. Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum error correction via codes over \(GF(4)\). IEEE Trans. Inform. Theory 44(4), 1369–1387 (1998)

    Article  MathSciNet  Google Scholar 

  7. Cavalcante, R.G., Lazari, H., Lima, J.D., Palazzo, R., Jr.: A new approach to the design of digital communication systems. In Discrete Mathematics and Theoretical Computer Science - DIMACS Series, Editors A. Ashikhimin and A. Barg, American Mathematical Society, 68:145–177, 2005

  8. de Albuquerque, C.D., Palazzo, R., Jr., da Silva, E.B.: New classes of topological quantum codes associated with self-dual, quasi self-dual and denser tessellations. Quantum Inf. Comput. 10(11 & 12), 0956–0970 (2010)

    MathSciNet  MATH  Google Scholar 

  9. de Albuquerque, C.D., Palazzo, R., Jr., da Silva, E.B.: Families of classes of topological quantum codes from tessellations \(\{4i+2,2i+1\},\{4i,4i\},\{8i-4,4\}\) and \(\{12i-6,3\}\). Quantum Inf. Comput. 14(15 & 16), 1424–1440 (2014)

    MathSciNet  Google Scholar 

  10. Delfosse, N.: Tradeoffs for reliable quantum information storage in surface codes and color codes. e-print arXiv:1301.6588

  11. Dennis, E., Kitaev, A., Landahl, A., Preskill, J.: Topological quantum memory. J. Math. Phys. 43, 4452 (2002)

    Article  ADS  MathSciNet  Google Scholar 

  12. Evans, Z.W.E., Stephens, A.M., Cole, J.H., Hollenberg, L.C.L.: Error correction optimisation in the presence of \(x/z\) asymmetry. e-print arXiv:0709.3875

  13. Ezerman, M.F., Ling, S., Solé, P.: Additive asymmetric quantum codes. IEEE Trans. Inform. Theory 57(8), 5536–5550 (2010)

    Article  MathSciNet  Google Scholar 

  14. Firby, P.A., Gardiner, C.F.: Surface Topology. Ellis Horwood Series in Mathematics and Its Applications (1991)

  15. Fujii, K., Tokunaga, Y.: Error-and-loss-tolerances of surface codes with general lattice structures. Phys. Rev. A 86, 020303 (2012)

    Article  ADS  Google Scholar 

  16. Hansen, J.P.: Toric surfaces, linear and quantum codes, secret sharing and decoding. e-print arXiv:1808.06487

  17. Ioffe, L., Mezard, M.: Asymmetric quantum error-correcting codes. Phys. Rev. A 75, 032345 (2007)

    Article  ADS  MathSciNet  Google Scholar 

  18. Ketkar, A., Klappenecker, A., Kumar, S., Sarvepalli, P.K.: Nonbinary stabilizer codes over finite fields. IEEE Trans. Inform. Theory 52(11), 4892–4914 (2006)

    Article  MathSciNet  Google Scholar 

  19. Kitaev, A.Y.: Quantum error correction with imperfect gates. In: Horita, O., Holevo, A.S., Caves, C.M. (eds.) Proc. Third Intern. Conf. on Quantum Communication and Measurement, New York, Plenum (1997)

  20. Kitaev, A.Y.: Fault-tolerant quantum computation by anyons. Ann. Phys. 303(1), 2–30 (2003)

    Article  ADS  MathSciNet  Google Scholar 

  21. La Guardia, G.G.: New families of asymmetric quantum BCH codes. Quantum Inform. Comput. 11(3–4), 239–252 (2011)

    Article  MathSciNet  Google Scholar 

  22. La Guardia, G.G.: Asymmetric quantum Reed–Solomon and generalized Reed–Solomon codes. Quantum Inform. Process. 11, 591–604 (2012)

    Article  MathSciNet  Google Scholar 

  23. La Guardia, G.G.: Asymmetric quantum product codes. Int. J. Quantum Inform. 10(1), 1250005(1–11) (2012)

  24. La Guardia, G.G.: Asymmetric quantum codes: new codes from old. Quantum Inform. Process. 12, 2771–2790 (2013)

    Article  ADS  MathSciNet  Google Scholar 

  25. Lidar, D.A., Brun, T.A.: Quantum Error Correction. Cambridge University Press, Cambridge (2013)

    Book  Google Scholar 

  26. Lima, E.L.: Homologia Básica (in portuguese). Editora do Instituto de Matemática Pura e Aplicada (IMPA) (2012)

  27. Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)

    MATH  Google Scholar 

  28. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum LDPC codes. In: Proc. Int. Symp. Inform. Theory (ISIT), pp. 305–309 (2008)

  29. Sarvepalli, P.K., Klappenecker, A., Rötteler, M.: Asymmetric quantum codes: constructions, bounds and performance. In: Proc. of the Royal Society A, pp. 1645–1672 (2009)

  30. Silva, E.B., Firer, M., Costa, S.R., Palazzo, R., Jr.: Signal constellations in the hyperbolic plane. J. Frankl. Inst. 343, 69 (2006)

    Article  Google Scholar 

  31. Steane, A.M.: Simple quantum error correcting-codes. Phys. Rev. A 54, 4741–4751 (1996)

    Article  ADS  MathSciNet  Google Scholar 

  32. Stephens, A.M., Evans, Z.W.E., Devitt, S.J., Hollenberg, L.C.L.: Asymmetric quantum error correction via code conversion. Phys. Rev. A 77, 062335 (2008)

    Article  ADS  MathSciNet  Google Scholar 

  33. Vick, J.W.: Homology Theory: An Introduction to Algebraic Topology. Springer, New York (1994)

    Book  Google Scholar 

  34. Wang, L., Feng, K., Ling, S., Xing, C.: Asymmetric quantum codes: characterization and constructions. IEEE Trans. Inform. Theory 56(6), 2938–2945 (2010)

    Article  MathSciNet  Google Scholar 

Download references

Acknowledgements

We would like to thank the Editor-in-Chief Yaakov S. Weinstein and the anonymous referees for their valuable comments and suggestions that helped to improve significantly the quality and the presentation of this paper. This research was supported in part by the Brazilian agency CNPq under Grant 305656/2015-5, 425224/2016-3 and 302759/2017-4. FAPEMIG under Grant APQ-00019-21.

Author information

Authors and Affiliations

  1. Science and Technology Center, Federal University of Cariri (UFCA), 63048-080, Juazeiro do Norte, CE, Brazil

    Clarice Dias de Albuquerque

  2. Department of Mathematics and Statistics, State University of Ponta Grossa (UEPG), 84030-900, Ponta Grossa, PR, Brazil

    Giuliano Gadioli La Guardia

  3. Department of Communications, FEEC-UNICAMP, 13083-852, Campinas, SP, Brazil

    Reginaldo Palazzo Jr.

  4. Department of Mathematics, Institute of Exact Sciences, Federal University of Alfenas (UNIFAL-MG), 37130-000, Alfenas, MG, Brazil

    Cátia Regina de Oliveira Quilles Queiroz

  5. Department of Mathematics, State University of Paraíba (UEPB), 58429-500, Campina Grande, PB, Brazil

    Vandenberg Lopes Vieira

Authors
  1. Clarice Dias de Albuquerque
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Giuliano Gadioli La Guardia
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Reginaldo Palazzo Jr.
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Cátia Regina de Oliveira Quilles Queiroz
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Vandenberg Lopes Vieira
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Clarice Dias de Albuquerque.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Albuquerque, C.D., La Guardia, G.G., Palazzo, R. et al. Euclidean and hyperbolic asymmetric topological quantum codes. Quantum Inf Process 21, 153 (2022). https://doi.org/10.1007/s11128-022-03488-8

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s11128-022-03488-8

Keywords

Search

Navigation