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arXiv:1401.3215 · IEEE International Symposium on Information Theory (ISIT)

Constructions of pure asymmetric quantum alternant codes based on subclasses of Alternant codes

Jihao Fan, Hanwu Chen

Published Jun 1, 2014• 1 citation
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Abstract

In this paper, we construct asymmetric quantum error-correcting codes(AQCs) based on subclasses of Alternant codes. Firstly, We propose a new subclass of Alternant codes which can attain the classical Gilbert-Varshamov bound to construct AQCs. It is shown that when dx= 2, Z-parts of the AQCs can attain the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous subclass of Alternant codes called Goppa codes. As an illustrative example, we get three [[55, 6, 19/4]], [[55, 10, 19/3]], [[55, 15, 19/2]] AQCs from the well known [55, 16, 19] binary Goppa code. At last, we get asymptotically good binary expansions of asymmetric quantum GRS codes, which are quantum generalizations of Retter's classical results. All the AQCs constructed in this paper are pure.

DOI
10.1109/isit.2014.6874962

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In this paper, we construct asymmetric quantum error-correcting codes(AQCs) based on subclasses of Alternant codes. Firstly, We propose a new subclass of Alternant codes which can attain the classical Gilbert-Varshamov bound to construct AQCs. It is shown that when dx= 2, Z-parts of the AQCs can attain the classical Gilbert-Varshamov bound. Then we construct AQCs based on a famous subclass of Alternant codes called Goppa codes. As an illustrative example, we get three [[55, 6, 19/4]], [[55, 10, 19/3]], [[55, 15, 19/2]] AQCs from the well known [55, 16, 19] binary Goppa code. At last, we get asymptotically good binary expansions of asymmetric quantum GRS codes, which are quantum generalizations of Retter's classical results. All the AQCs constructed in this paper are pure.
The stabilizer method for constructing a class of asymmetric quantum codes (AQC), called additive AQC, has been established by Aly et.al. In this paper, we present a new characterization of AQC, which generalizes a result of the symmetric case known previously. As an application of the characterization, we establish a relationship of AQC with classical error-correcting codes and show a few examples of good AQC with specific parameters. By using this relationship, we obtain an asymptotic bound on AQCs from algebraic geometry codes.
Since Calderbank et al.established CRSS construction,which constructs quantum error-correcting codes(QECC) from classical error-correcting codes,many QECC demonstrating good performance have been constructed from classical error-correcting codes;these codes are referred to as quantum stabilizer codes.Recent physical experiments indicate that the probabilities of occurrence of qubit-flip errors are much less than those of phase-flip errors in most quantum mechanical systems.Error-correcting codes thus constructed are referred to as asymmetric quantum codes.In this paper,series of asymmetric quantum stabilizer codes are constructed using nested Goppa codes and dual contained Goppa codes.In the construction of asymmetric quantum codes based on nested Goppa codes,the choice of Goppa codes is limited to construct codes with analysis methods.In normal circumstances,the construction is executed with the help of mathematical software Matlab,which is used to compute the minimum distance of Goppa codes.In the construction of asymmetric quantum codes based on dual contained Goppa codes,the error correction ability of the codes constructed in this paper is reflected in the correction of Z-type errors.
The concept of asymmetric entanglement-assisted quantum error-correcting code (asymmetric EAQECC) is introduced in this article. Codes of this type take advantage of the asymmetry in quantum errors since phase-shift errors are more probable than qudit-flip errors. Moreover, they use pre-shared entanglement between encoder and decoder to simplify the theory of quantum error correction and increase the communication capacity. Thus, asymmetric EAQECCs can be constructed from any pair of classical linear codes over an arbitrary field. Their parameters are described and a Gilbert-Varshamov bound is presented. Explicit parameters of asymmetric EAQECCs from BCH codes are computed and examples exceeding the introduced Gilbert-Varshamov bound are shown.
Asymmetric quantum error-correcting codes (AQCs) may offer some advantage over their symmetric counterparts by providing better error-correction for the more frequent error types. The well-known CSS construction ofqqq-ary AQCs is extended by removing the$ \BBF _{q}−linearityrequirementaswellasthelimitationonthetypeofinnerproductused.TheproposedconstructionsarecalledCSS−likeconstructionsandutilizepairsofnestedsubfieldlinearcodesunderoneoftheEuclidean,traceEuclidean,Hermitian,andtraceHermitianinnerproducts.Afterestablishingsometheoreticalfoundations,best−performingCSS−likeAQCsareconstructed.Combiningsomeconstructionsofnestedpairsofclassicalcodesandlinearprogramming,manyoptimalandgoodpure-linearity requirement as well as the limitation on the type of inner product used. The proposed constructions are called CSS-like constructions and utilize pairs of nested subfield linear codes under one of the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. After establishing some theoretical foundations, best-performing CSS-like AQCs are constructed. Combining some constructions of nested pairs of classical codes and linear programming, many optimal and good pure−linearityrequirementaswellasthelimitationonthetypeofinnerproductused.TheproposedconstructionsarecalledCSS−likeconstructionsandutilizepairsofnestedsubfieldlinearcodesunderoneoftheEuclidean,traceEuclidean,Hermitian,andtraceHermitianinnerproducts.Afterestablishingsometheoreticalfoundations,best−performingCSS−likeAQCsareconstructed.Combiningsomeconstructionsofnestedpairsofclassicalcodesandlinearprogramming,manyoptimalandgoodpureq−aryCSS−likecodesfor-ary CSS-like codes for−aryCSS−likecodesforq \in \{ 2,3,4,5,7,8,9\}uptoreasonablelengthsarefound.Inmanyinstances,removingtheup to reasonable lengths are found. In many instances, removing theuptoreasonablelengthsarefound.Inmanyinstances,removingthe \BBF _{q}$-linearity and using alternative inner products give us pure AQCs with improved parameters than relying solely on the standard CSS construction.