nLab stabilizer code

Idea

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.

Properties

Relation to classical binary 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)

References

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:

• Simeon Ball, Aina Centelles, Felix Huber, Section 2 of: Quantum error-correcting codes and their geometries (arXiv:2007.05992)

See also

• Wikipedia, Stabilizer code

Realization in experiment:

Realization of quantum error correction in experiment:

• D. G. Cory, M. D. Price, W. Maas, Emanuel Knill, Raymond Laflamme, Wojchiek H. Zurek, T. F. Havel, and S. S. Somaroo, Experimental Quantum Error Correction, Phys. Rev. Lett. 81, 2152 (1998) (doi:10.1103/PhysRevLett.81.2152)