Clifford Codes

4 Clifford Codes

A fundamental problem is to decide when a Clifford codes is not equal to a stabilizer code. If we are given a Clifford code defined by $(N,χ )$ with nonabelian normal subgroup $N$ of $E$ , there might exist an abelian normal subgroup $N0$ of $E$ and a one-dimensional character of $N0$ so that the two codes are the same,

∑ ∑ P = χ(1) χ(n-1)ρ(n) = χ(1) χ0(n-1)ρ(n) = P0. |N| n∈N |N0 |n∈N0

In other words, the Clifford code might turn out to be a stabilizer code. The publications below give criteria that allow one to decide this question.

A. Klappenecker and M. Rötteler. Remarks on Clifford codes. In IEEE International Symposium on Information Theory, page 354, 2004

@InProceedings{klappenecker044,
  author = {Klappenecker, A. and R{\"o}tteler, M.},
  title = {Remarks on {C}lifford Codes},
  booktitle = {IEEE International Symposium on Information Theory},
  pages = {354},
  year = {2004}
}

A. Klappenecker and M. Rötteler. On the structure of nonstabilizer Clifford codes. Quantum Information and Computation, 4(2):152–160, 2004

@Article{klappenecker045,
  author = {Klappenecker, A. and R\"otteler, M.},
  title = {On the Structure of Nonstabilizer {C}lifford Codes},
  journal = {Quantum Information and Computation},
  year = 2004,
  volume = 4,
  number = 2,
  pages = {152-160}
}

Efficient Decoherence Control Algorithms

4 Clifford Codes