A Construction of Quantum Stabilizer Codes Based on Syndrome Assignment by Classical Parity-Check Matrices.

In this paper, a new but simple construction of stabilizer codes and related entanglement-assisted quantum error-correcting codes is proposed based on syndrome assignment by classical parity-check matrices. This method turns the construction of quantum stabilizer codes to the construction of classical parity-check matrices satisfying a specific commutative condition. The designed minimum distance 2t^*+1 of the constructed quantum stabilizer codes can be achieved by a commutative classical parity-check matrix with classical minimum distance 4t^*-m, where the parameter m, 0\leq m\leq 2t^*, depends on a property of the parity-check matrix. As m decreases, there is an increasing set of additional correctable error operators beyond the designed error correcting capability t^*. The (asymptotic) coding efficiency is at least comparable to that of CSS codes. A class of quantum Reed–Muller codes is constructed and codes in this class have a larger set of correctable error operators than that of the quantum Reed–Muller codes previously developed in the literature. Quantum circulant codes are also constructed and many of them are optimal in terms of their coding parameters. [ABSTRACT FROM AUTHOR]