Fault-Tolerant Quantum Computation with Higher-Dimensional Systems

  • Daniel Gottesman
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 1509)


Instead of a quantum computer where the fundamental units are 2-dimensional qubits, we can consider a quantum computer made up of d-dimensional systems. There is a straightforward generalization of the class of stabilizer codes to d-dimensional systems, and I will discuss the theory of fault-tolerant computation using such codes. I prove that universal fault-tolerant computation is possible with any higher-dimensional stabilizer code for prime d.


Quantum Code Quantum Error Correction Stabilizer Code Pauli Group Error Syndrome 
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.
    P. Shor, Phys. Rev. A 52, 2493 (1995).Google Scholar
  2. 2.
    A. M. Steane, Phys. Rev. Lett. 77, 793 (1996).zbMATHCrossRefMathSciNetGoogle Scholar
  3. 3.
    D. Gottesman, Phys. Rev. A 54, 1862 (1996).MathSciNetGoogle Scholar
  4. 4.
    A.R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, Phys. Rev.Lett. 78, 405 (1997).zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    A. R. Calderbank, E. M. Rains, P. W. Shor, and N. J. A. Sloane, “Quantum error correction via codes over GF(4),„ quant-ph/9608006, to appear in IEEE Trans. Information Theory.Google Scholar
  6. 6.
    P. Shor, Proceedings of the 37th Symposium on the Foundations of Computer Science, IEEE Computer Society Press (Los Alamitos, CA), 56 (1996).Google Scholar
  7. 7.
    D. Gottesman, Phys. Rev. A 57, 127 (1998).CrossRefGoogle Scholar
  8. 8.
    E. Knill, “Non-binary error bases and quantum codes,„ quant-ph/9608048.Google Scholar
  9. 9.
    E. Knill, “Group representations, error bases and quantum codes,„ quant-ph/9608049.Google Scholar
  10. 10.
    E. Rains, “Nonbinary quantum codes,„ quant-ph/9703048.Google Scholar
  11. 11.
    D. DiVincenzo and P. Shor, Phys. Rev. Lett. 77, 3260 (1996).CrossRefGoogle Scholar
  12. 12.
    E. Knill and R. Laflamme, private communication.Google Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 1999

Authors and Affiliations

  • Daniel Gottesman
    • 1
  1. 1.T-6 GroupLos Alamos National LaboratoryUSA

Personalised recommendations