Stabilizer code

An ${\displaystyle [n,k]}$ stabilizer code is a subspace ${\displaystyle V_{S}}$ of the Hilbert space of n qubits ${\displaystyle (\mathbb {C} ^{2})^{\otimes n}}$, which is stabilized by an abelian subgroup ${\displaystyle S}$ of the Pauli group on n qubits ${\displaystyle G_{n}}$ such that ${\displaystyle -I\notin S}$ and ${\displaystyle S}$ has ${\displaystyle n-k}$ independent generators, ${\displaystyle S=\langle g_{1},\ldots ,g_{n-k}\rangle }$. These conditions are required so that the vector space stabilized is non-trivial. Denote this code ${\displaystyle C(S)}$.

References

• D. Gottesman, "Stabilizer codes and quantum error correction," quant-ph/9705052, Caltech Ph.D. thesis. http://arxiv.org/abs/quant-ph/9705052