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arXiv:2502.03242

Constructions and decoding procedures for quantum CSS codes

Yannick Saouter, Massinissa Zenia, Gilles Burel

Published Feb 5, 2025• 0 citations
arXivHTMLPDF

Abstract

This article presents new constructions of quantum error correcting Calderbank-Shor-Steane (CSS for short) codes. These codes are mainly obtained by Sloane's classical combinations of linear codes applied here to the case of self-orthogonal linear codes. A new algebraic decoding technique is also introduced. This technique is exemplified on CSS codes obtained from BCH, Reed-Muller and projective geometry codes.

DOI
10.48550/arxiv.2502.03242

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This article presents new constructions of quantum error correcting Calderbank-Shor-Steane (CSS for short) codes. These codes are mainly obtained by Sloane's classical combinations of linear codes applied here to the case of self-orthogonal linear codes. A new algebraic decoding technique is also introduced. This technique is exemplified on CSS codes obtained from BCH, Reed-Muller and projective geometry codes.
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.
Quantum error-correcting codes are essential for protecting quantum information against decoherence. This work investigates the construction of quantum CSS codes from classical self-orthogonal binary linear codes derived from extended quadratic residue codes. By exploiting the large automorphism groups of these codes, the minimum-weight codewords are partitioned according to the action of maximal subgroups of the automorphism group. Codes generated from these partitions are shown to be self-orthogonal and to retain large automorphism groups, allowing the construction of associated quantum CSS codes with good parameters. The resulting codes include new quantum codes with lengths ranging from 16 to 48 and minimum distances up to 11. Their automorphism groups and permutation-decoding properties are also analyzed, including the number of permutations required for complete decoding. The results provide new families of quantum CSS codes combining good error-correcting parameters, large automorphism groups, and efficient permutation decoding.
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.
This thesis proposes, as the main contributions, constructions method of new families of quantum CSS codes.These constructions are based on classical cyclic codes of the types BCH, Reed-Solomon, Reed-Muller, Quadratic Residue and also are based on product codes of classical Reed-Solomon codes.The main family of quantum codes constructed in this work, i. e., quantum codes having better parameters, are the ones derived from classical BCH codes.Moreover, we present some new conditions in which quantum CSS cyclic codes are quantum MDS codes.In addition, we provide the elements to connect matroid theory and quantum coding theory.More specifically, we show that the weight enumerator of a CSS quantum code is equivalent to evaluating the Tutte polynomial of the direct sum of the matroid associated to the classical codes used in the CSS construction.