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arXiv:quant-ph/0608063 · IEICE Technical Report; IEICE Tech. Rep.

Several Classes of Concatenated Quantum Codes: Constructions and Bounds

Hachiro Fujita

Published Aug 8, 2006• 3 citations
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Abstract

In this paper we present several classes of asymptotically good concatenated quantum codes and derive lower bounds on the minimum distance and rate of the codes. We compare these bounds with the best-known bound of Ashikhmin--Litsyn--Tsfasman and Matsumoto. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.

DOI
10.48550/arxiv.quant-ph/0608063

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In this paper we present several classes of asymptotically good concatenated quantum codes and derive lower bounds on the minimum distance and rate of the codes. We compare these bounds with the best-known bound of Ashikhmin--Litsyn--Tsfasman and Matsumoto. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.
In classical coding theory code concatenation is successfully used to construct good errorcorrecting codes and most of the asymptotically good codes known so far are based on concatenation. In this paper we present some classes of asymptotically good concatenated quantum codes, which are a quantum analogue of classical concatenated codes, and derive lower bounds on the minimum distance and the rate of the codes. Our bounds improve on the best lower bound of Ashikhmin–Litsyn–Tsfasman and Matsumoto for rates smaller than about one half. We also give a polynomial-time decoding algorithm for the codes that can decode up to one fourth of the lower bound on the minimum distance of the codes.
It is known that quantum error correction can be achieved using classical binary codes or additive codes over F/sub 4/. Asymptotically good quantum codes have been constructed from algebraic-geometry codes and a bound on (/spl delta/, R) was computed from the Tsfasman-Vladut-Zink bound of the theory of classical algebraic-geometry codes. In this correspondence, by the use of a concatenation technique we construct a family of asymptotically good quantum codes exceeding the bound in a small interval.
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.
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.