On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction

  • Gilles Zémor
Part of the Lecture Notes in Computer Science book series (LNCS, volume 5557)


We review constructions of quantum surface codes and give an alternative, algebraic, construction of the known classes of surface codes that have fixed rate and growing minimum distance. This construction borrows from Margulis’s family of Cayley graphs with large girths, and highlights the analogy between quantum surface codes and cycle codes of graphs in the classical case. We also attempt a brief foray into the class of quantum topological codes arising from higher dimensional manifolds and find these examples to have the same constraint on the rate and minimum distance as in the 2-dimensional case.


Minimum Distance Regular Graph Cayley Graph Dual Graph Quantum Code 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Alon, N., Hoory, S., Linial, N.: The Moore bound for irregular graphs. Graphs Combin. 18, 53–57Google Scholar
  2. 2.
    Bombin, H., Martin-Delgado, M.A.: Homological Error Correction: Classical and Quantum Codes. J. Math. Phys. 48, 052105 (2007)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Calderbank, A.R., Shor, P.W.: Good quantum error-correcting codes exist. Phys. Rev. A 54, 1098 (1996)CrossRefGoogle Scholar
  4. 4.
    Decreusefond, L., Zémor, G.: On the error-correcting capabilities of cycle codes of graphs. Combinatorics, Probability and Computing 6, 27–38 (1997)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Dennis, E., Kitaev, A., Landahl, A., Preskill, J.: Topological quantum memory. J. Math. Phys. 43, 4452–4505 (2002)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Erdös, P., Sachs, H.: Reguläre Graphen gegebener Taillenweite mit minimaler Knotenzahl. Wiss. Z. Martin-Luther-Univ. Halle-Wittenberg, Math.-Naturwiss. Reihe 12, 251–258 (1963)Google Scholar
  7. 7.
    Freedman, M.H., Meyer, D.A., Luo, F.: F2-systolic freedom and quantum codes. In: Mathematics of quantum computation. Comput. Math. Ser., pp. 287–320. Chapman & Hall/CRC, Boca Raton (2002)Google Scholar
  8. 8.
    Hagiwara, M., Imai, H.: Quantum Quasi-Cyclic LDPC Codes. In: Proc. IEEE International Symposium Information Theory (ISIT), Nice 2007, pp. 806–810 (2007)Google Scholar
  9. 9.
    Hatcher, A.: Algebraic Topology. Cambridge University Press, Cambridge (2002)zbMATHGoogle Scholar
  10. 10.
    Kim, I.H.: Quantum codes on Hurwitz surfaces, S. B. Thesis, MIT (2007),
  11. 11.
    Kitaev, A.: Quantum error correction with imperfect gates. In: Proc. 3rd nt. Conf. of Quantum Communication and Measurement (1997)Google Scholar
  12. 12.
    Mackay, D.J.C., Mitchison, G., Mcfadden, P.L.: Sparse Graph Codes for Quantum Error-Correction. IEEE Trans. Inform. Theory 50(10), 2315–2330 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  13. 13.
    Margulis, G.A.: Explicit constructions of graphs without short cycles and low density codes. Combinatorica 2(1), 71–78 (1982)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Preskill, J.: Quantum computation,
  15. 15.
    Širáň, J.: Triangle group representations and constructions of regular maps. Proc. London Math. Soc. 82(3), 513–532 (2001)MathSciNetCrossRefzbMATHGoogle Scholar
  16. 16.
    Steane, A.: Multiple particle interference and quantum error correction. Proc. Roy. Soc. Lond. A 452, 2551 (1996)MathSciNetCrossRefzbMATHGoogle Scholar
  17. 17.
    Takeda, K., Nishimori, H.: Self-dual random-plaquette gauge model and the quantum toric code. Nuclear Physics B 686(3), 377–396 (2004)MathSciNetCrossRefzbMATHGoogle Scholar
  18. 18.
    Tillich, J.-P., Zémor, G.: Optimal cycle codes constructed from Ramanujan graphs. Siam J. on Discrete Math. 10(3), 447–459 (1997)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2009

Authors and Affiliations

  • Gilles Zémor
    • 1
  1. 1.Institut de Mathématiques de BordeauxUMR 5251, Université Bordeaux 1TalenceFrance

Personalised recommendations