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IEEE International Mediterranean Conference on Communications and Networking (MeditCom)

Algebraic decoding for classes of quantum codes

Yannick Saouter, Gilles Burel

Published Jul 8, 2024• 0 citations
Publisher

Abstract

This paper describes a technique to perform the decoding of quantum Calderbank-Steane-Shor codes built from self-orthogonal classical binary codes possessing algebraic decoding procedures. This technique gives a viable alternative to syndrome decoding for codes with large minimum distance. The cases of Reed-Muller and BCH codes are examined. The new method is compared with syndrome decoding on illustrative examples. The asymptotic cost of both methods is also given. Index Terms-quantum code, algebraic decoding, CSS quantum codes.

DOI
10.1109/meditcom61057.2024.10621269

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This paper describes a technique to perform the decoding of quantum Calderbank-Steane-Shor codes built from self-orthogonal classical binary codes possessing algebraic decoding procedures. This technique gives a viable alternative to syndrome decoding for codes with large minimum distance. The cases of Reed-Muller and BCH codes are examined. The new method is compared with syndrome decoding on illustrative examples. The asymptotic cost of both methods is also given. Index Terms-quantum code, algebraic decoding, CSS quantum codes.
In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.
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.
It is an important issue to search for self-orthogonal codes for construction of quantum codes by CSS construction (Calderbank-Sho-Steane codes); in quantum error correction, CSS codes are a special type of stabilizer codes constructed from classical codes with some special properties, and the CSS construction of quantum codes is a well-known construction. First, we employ hierarchical posets with two levels for construction of binary linear codes. Second, we find some necessary and sufficient conditions for these linear codes constructed using posets to be self-orthogonal, and we use these self-orthogonal codes for obtaining binary quantum codes. Finally, we obtain four infinite families of binary quantum codes for which the minimum distances are three or four by CSS construction, which include binary quantum Hamming codes with length n≥7. We also find some (almost) “optimal” quantum codes according to the current database of Grassl. Furthermore, we explicitly determine the weight distributions of these linear codes constructed using posets, and we present two infinite families of some optimal binary linear codes with respect to the Griesmer bound and a class of binary Hamming codes.
The polynomial representation of a quantum state is defined.Based on the Calderbank-Shor-Steane(CSS)-type construction of quantum codes,a new method for encoding and decoding of CSS-type quantum convolution codes is presented.Its corresponding networks are also described.The basis state of the code is transformed into the multiplication of an information polynomial by the generator polynomial.Then networks can be realized with operations of polynomial multiplication. The simple method possesses a highly regular structure,and the networks are easy to realize.