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Concatenation of Quantum Error-Correcting Codes and Lower Bounds on the Largest Minimum Distance

Hamada Mitsuru

Published Jul 18, 2008• 0 citations
OpenAlex

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A method for concatenating quantum error-correcting codes is presented. The method is applicable to a wide class of quantum error-correcting codes known as Calderbank-Shor-Steane (CSS) codes. As a result, codes that achieve a high rate in the Shannon-theoretic sense and that are decodable in polynomial time are presented. The rate is the highest among those known to be achievable by CSS codes. Moreover, the best known lower bound on the greatest minimum distance of codes constructible in polynomial time is improved for a wide range.
Recently, G. Zemor (see IEEE Trans. Inf. Theory, vol.47, p.835-7, 2001) proposed an improvement on the Sipser-Spielman analysis of expander codes (Sipser, M. and Spielman, D.A., IEEE Trans. Inf. Theory, vol.42 , p.1710-22, 1996) and presented a linear-time iterative decoder that can correct a number of errors up to approximately 1/4 the known lower bound on the minimum distance of the code. We propose an improvement on Zemor's decoder for F=GF(2), with the number of correctable errors becoming close to half the lower bound on the minimum distance. The improvement is obtained by inserting into the decoding algorithm features akin to generalized minimum distance decoding of concatenated codes.