In quantum information theory, a stabilizer code is a quantum error correcting code whose code subspace is the fixed subspace of an abelian subgroup $\{-1\} \notin S \subset \mathcal{P}_n$ of the Pauli group $\mathcal{P}_n$ acting on a register of $n$ q-bits.
Conversely, $S \subset \mathcal{P}_n$ is then the stabilizer subgroup of the code space, whence the name of this class of codes.
Most quantum error correcting codes known are in fact stabilizer codes.
Quantum stabilizer codes are closely related to classical error correcting codes, specifically to binary linear codes.
(e.g. Ball, Centelles & Huber 20, Sec. 2.3)
Stabilizer codes were introduced, independeny, in
Daniel Gottesman, Stabilizer Codes and Quantum Error Correction (arXiv:quant-ph/9705052)
Robert Calderbank, E. M Rains, Peter W. Shor, N. J. A. Sloane, Quantum Error Correction and Orthogonal Geometry, Phys. Rev. Lett. 78:405-408, 1997 (arXiv:quant-ph/9605005)
following
Peter W. Shor, Scheme for reducing decoherence in quantum computer memory, Phys. Rev. A 52, R2493(R) 1995 (doi:10.1103/PhysRevA.52.R2493)
Andrew M. Steane, Multiple Particle Interference and Quantum Error Correction, Proc. Roy. Soc. Lond. A452 (1996) 2551 (arXiv:quant-ph/9601029)
Review:
See also
Realization in experiment:
Realization of quantum error correction in experiment: