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arXiv:1006.1694 · International Journal of Quantum Information

PURE ASYMMETRIC QUANTUM MDS CODES FROM CSS CONSTRUCTION: A COMPLETE CHARACTERIZATION

Martianus Frederic Ezerman, SOMPHONG JITMAN, HAN MAO KIAH, SAN LING

Published Apr 1, 2013• 30 citations
arXivHTMLPDF

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 maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.

DOI
10.1142/s0219749913500275

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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 maximum distance separable (MDS) Conjecture, pure CSS AQMDS codes of all possible parameters are accounted for.
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.
This thesis focuses on the construction and analysis of the parameters of asymmetric quantum error-correcting codes (AQECCs). We use the so-called functional approach to connect AQECCs to classical codes via orthogonal arrays. Based on the functional approach, we derive pure AQECCs from CSS-like constructions that include the standard CSS construction known prior to this work. The CSS-like constructions remove both the specific choice of Euclidean inner product and the $\F_{q}$-linearity condition imposed by the standard variant. Hence, pure AQECCs can now be constructed by using pairs of nested $\F_{r}$-linear codes over $\F_{q}$ where $\F_{r}$ is any subfield of $\F_{q}$ under the Euclidean, trace Euclidean, Hermitian, and trace Hermitian inner products. Relationships between the various CSS-like constructions are also exhibited. A formal definition of asymmetric stabilizer codes is given and a connection between pure asymmetric stabilizer codes and AQECCs derived from the CSS-like constructions is established. It is shown that the class of pure CSS-like AQECCs forms a subset of asymmetric stabilizer codes. We show how classical linear MDS codes can be used to construct AQECCs that satisfy the quantum Singleton bound with dz≥dx≥2d_{z} \geq d_{x}\geq 2dz​≥dx​≥2 for all possible values of length nnn for which linear MDS codes over $\F_{q}$ are known to exist. Beyond the lengths specified by the classical MDS conjecture, various explicit constructions of nested pairs of classical codes can be combined with linear programming to establish the optimality of pure CSS-like codes with parameters [[n,k,dz/dx]]q[[n,k,d_{z}/d_{x}]]_{q}[[n,k,dz​/dx​]]q​. For q={2,3,4,5,7,8,9}q=\{2,3,4,5,7,8,9\}q={2,3,4,5,7,8,9}, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented.
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.
The asymmetric CSS construction is extended to the Hermitian dual case. New infinite families of quantum symmetric and asymmetric codes are constructed. In particular, new quantum codes are obtained from binary BCH codes and MDS codes. These codes have known minimum distances and thus the relationship between the rate gain and minimum distance is given explicitly. The codes obtained are shown to have parameters better than those of previous codes. A number of known codes are special cases of the codes given here.
Abstract : We consider quantum error-correcting codes over alphabets of arbitrary size. We generalize quantum stabilizer codes and develop Calderbank-Shor-Steane construction over quantum systems of arbitrary dimensions using the group structure of alphabets. We also develop a methodology systematically conjoining error-correcting codes into a new class of error-correcting codes. With the help of these methods we give several ways to construct quantum maximum distance separable (MDS) codes and present many families of quantum MDS codes.