Variations on Encoding Circuits for Stabilizer Quantum Codes

  • Markus Grassl
Part of the Lecture Notes in Computer Science book series (LNCS, volume 6639)


Quantum error-correcting codes (QECC) are an important component of any future quantum computing device. After a brief introduction to stabilizer quantum codes, we present two methods to efficiently compute encoding circuits for them.


Quantum Circuit Quantum Code Tensor Factor Stabilizer Matrix Quantum Error Correction 
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.


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  1. 1.
    Ashikhmin, A., Knill, E.: Nonbinary quantum stabilizer codes. IEEE Transactions on Information Theory 47(7), 3065–3072 (2001) (preprint quant-ph/0005008)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Calderbank, A.R., Rains, E.M., Shor, P.W., Sloane, N.J.A.: Quantum Error Correction Via Codes over GF(4). IEEE Transactions on Information Theory 44(4), 1369–1387 (1998) (preprint quant-ph/9608006)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Cleve, R., Gottesman, D.: Efficient computations of encodings for quantum error correction. Physical Review A 56(1), 76–82 (1997)CrossRefGoogle Scholar
  4. 4.
    Gottesman, D.: A Class of Quantum Error-Correcting Codes Saturating the Quantum Hamming Bound. Physical Review A 54(3), 1862–1868 (1996)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Gottesman, D.: Stabilizer Codes and Quantum Error Correction. PhD thesis, California Institute of Technology, Pasadena, California (1997)Google Scholar
  6. 6.
    Gottesman, D.: Errata in “Stabilizer Codes and Quantum Error Correction” (2004),
  7. 7.
    Grassl, M.: Algorithmic aspects of quantum error-correcting codes. In: Brylinski, R.K., Chen, G. (eds.) Mathematics of Quantum Computation, pp. 223–252. CRC Press, Boca Raton (2002)Google Scholar
  8. 8.
    Grassl, M., Klappenecker, A., Rötteler, M.: Graphs, Quadratic Forms, and Quantum Codes. In: Proceedings of the 2002 IEEE International Symposium on Information Theory, June 30 - July 5, p. 45. IEEE, Los Alamitos (2002) (preprint quant-ph/0703112)Google Scholar
  9. 9.
    Grassl, M., Rötteler, M., Beth, T.: Efficient Quantum Circuits for Non-Qubit Quantum Error-Correcting Codes. International Journal of Foundations of Computer Science (IJFCS) 14(5), 757–775 (2003) (preprint quant-ph/0211014)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Grassl, M., Rötteler, M., Beth, T.: On Quantum MDS codes. In: Proceedings of the 2004 IEEE International Symposium on Information Theory, Chicago, June 25 - July 2, p. 355. IEEE, Los Alamitos (2004)Google Scholar
  11. 11.
    Knill, E., Laflamme, R.: Theory of quantum error-correcting codes. Physical Review A 55(2), 900–911 (1997) (preprint quant-ph/9604034)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Kuo, K.Y., Lu, C.C.: A Further Study on the Encoding Complexity of Quantum Stabilizer Codes. In: Proceedings 2010 International Symposium on Information Theory and its Applications (ISITA), October 17-20, pp. 1041–1044 (2010)Google Scholar
  13. 13.
    Nielsen, M.A., Chuang, I.L.: Quantum Computation and Quantum Information. Cambridge University Press, Cambridge (2000)zbMATHGoogle Scholar
  14. 14.
    Schlingemann, D.: Stabilizer codes can be realized as graph codes. Quantum Information & Computation 2(4), 307323 (2002) (preprint quant-ph/0111080)MathSciNetzbMATHGoogle Scholar
  15. 15.
    Schlingemann, D., Werner, R.F.: Quantum error-correcting codes associated with graphs. Physical Review A 65(1), 012308 (2002) (preprint quant-ph/0012111)CrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2011

Authors and Affiliations

  • Markus Grassl
    • 1
  1. 1.Centre for Quantum TechnologiesNational University of SingaporeSingaporeRepublic of Singapore

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