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

Publisher:
IEEE-Inst Electrical Electronics Engineers Inc
Publication Type:
Journal Article
Citation:
IEEE Transactions On Information Theory, 2011, 57 (10), pp. 7163 - 7179
Issue Date:
2011-01
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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 = m = 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.
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