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Pure Asymmetric Quantum MDS Codes from CSS Construction: A Complete Characterization

Yeow Meng Chee, Somphong Jitman, Han Mao Kiah, San Ling, Martianus Frederic Ezerman

Published Aug 15, 2016• 35 citations
OpenAlex

Abstract

Using the Calderbank-Shor-Steane (CSS) construction, pure $q$-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

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In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length 6ps over Fpm.We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 6ps over Fpm. Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.

IEEE Access•Jan 1, 2021•7 citations
Using the Calderbank-Shor-Steane (CSS) construction, pure qqq-ary asymmetric quantum error-correcting codes attaining the quantum Singleton bound are constructed. Such codes are called pure CSS asymmetric quantum maximum distance separable (AQMDS) codes. Assuming the validity of the classical MDS Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.
Recently, Galindo et al. introduced the concept of asymmetric entanglement-assisted quantum error-correcting codes (AEAQECCs) from Calderbank-Shor-Steane (CSS) construction. In general, it's difficult to determine the required number of maximally entangled states of an AEAQECC, which is associated with the dimension of the intersection of the two corresponding linear codes. Two linear codes are said to be a linear l-intersection pair if their intersection has dimension l. In this paper, all possible linear l-intersection pairs of MDS codes are given. As an application, we give a complete characterization of pure MDS AEAQECCs for all possible parameters.
Due to the asymmetry of quantum errors, phase-shift errors are more likely to occur than qubit-flip errors. Consequently, there is a need to develop asymmetric quantum error-correcting (QEC) codes that can safeguard quantum information transmitted through asymmetric channels. Currently, a significant body of literature has investigated the construction of asymmetric QEC codes. However, the asymmetry of most QEC codes identified in the literature is limited by the dual-containing condition within the Calderbank-Shor-Steane (CSS) framework. This limitation restricts the exploration of their full potential in terms of asymmetry. In order to enhance the asymmetry of asymmetric QEC codes, we utilize entanglement-assisted technology and exploit the algebraic structure of cyclotomic cosets of constacyclic codes to achieve this goal. In this paper, we generalize the decomposition method of the defining set for constacyclic codes and apply it to count the number of pre-shared entangled states in order to construct four new classes of asymmetric entanglement-assisted quantum maximal-distance separable (EAQMDS) codes that satisfy the asymmetric entanglement-assisted quantum Singleton bound. Compared with the codes existing in the literature, the lengths of the constructed EAQMDS codes and the number of pre-shared entangled states are more general, and the codes constructed in this paper have greater asymmetry.
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.
Recently, considerable works of literature have grown up around the theme of quantum error-correcting codes. Quantum convolutional maximum distance separable (MDS for brevity) codes and MDS asymmetric quantum codes often have better properties in long-distance quantum communications and asymmetric quantum channels. Within this writing, the authors utilized constacyclic codes in constructing three new kinds of MDS quantum convolutional codes and a class of MDS asymmetric quantum codes. These results are new in comparison with those presented in the existing publications on this subject.
Quantum error correction is fundamentally important for quantum information processing and computation. Quantum error correction codes have been studied and constructed since the pioneering papers of Shor and Steane. Optimal (called MDS) qqq-qubit quantum codes attaining the quantum Singleton bound were constructed for very restricted lengths n≤q2+1n \leq q^2+1n≤q2+1. Entanglement-assisted quantum error correction (EAQEC) code was proposed to use the pre-shared maximally entangled state for the enhancing of error correction capability. Recently there have been a lot of constructions of MDS EAQEC codes attaining the quantum Singleton bound for very restricted lengths. In this paper we construct such MDS EAQEC [[n,k,d,c]]q[[n, k, d, c]]_q[[n,k,d,c]]q​ codes for arbitrary nnn satisfying n≤q2+1n \leq q^2+1n≤q2+1 and arbitrary distance d≤n+22d\leq \frac{n+2}{2}d≤2n+2​. It is proved that for any given length nnn satisfying O(q2)=n≤q2+1O(q^2)=n \leq q^2+1O(q2)=n≤q2+1 and any given distance ddd satisfying $ O(q^2)=d \leq \frac{n+2}{2}$, there exist at least O(q2)O(q^2)O(q2) MDS EAQEC [[n,k,d,c]]q[[n, k, d, c]]_q[[n,k,d,c]]q​ codes with different ccc parameters. Our results show that there are much more MDS entanglement-assisted quantum codes than MDS quantum codes without consumption of the maximally entangled state. This is natural from the physical point of view. Our method can also be applied to construct MDS entanglement-assisted quantum codes from the generalized MDS twisted Reed-Solomon codes.
In this paper, we use the CSS and Steane’s constructions to establish quantum error-correcting codes (briefly, QEC codes) from cyclic codes of length 6ps over Fpm.We obtain several new classes of QEC codes in the sense that their parameters are different from all the previous constructions. Among them, we identify all quantum MDS (briefly, qMDS) codes, i.e., optimal quantum codes with respect to the quantum Singleton bound. In addition, we construct quantum synchronizable codes (briefly, QSCs) from cyclic codes of length 6ps over Fpm. Furthermore, we give many new QSCs to enrich the variety of available QSCs. A lot of them are QSCs codes with shorter lengths and much larger minimum distances than known non-primitive narrow-sense BCH codes.