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 $d_{z} \geq d_{x}\geq 2$ for all possible values of length $n$ 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 $[[n,k,d_{z}/d_{x}]]_{q}$ . For $q=\{2,3,4,5,7,8,9\}$ , lists of optimal asymmetric CSS-like codes for reasonable lengths are presented.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 $d_{z} \geq d_{x}\geq 2$ for all possible values of length $n$ 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 $[[n,k,d_{z}/d_{x}]]_{q}$. For $q=\{2,3,4,5,7,8,9\}$, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented.

Jan 1, 20112 citations

The Correcting Capacities of Linear Codes

M. V. Kozlov

Soviet physics. DokladyNov 1, 19693 citations

Optimal Asymmetric Quantum Codes from the Euclidean Sums of Linear Codes

Peng Xu, Xiusheng Liu

In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.In this paper, we first give the definition of the Euclidean sums of linear codes, and prove that the Euclidean sums of linear codes are Euclidean dual-containing. Then we construct two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of the Reed-Solomon codes, and two new classes of optimal asymmetric quantum error-correcting codes based on Euclidean sums of linear codes generated by Vandermonde matrices over finite fields. Moreover, these optimal asymmetric quantum error-correcting codes constructed in this paper are different from the ones in the literature.

Wuhan University Journal of Natural SciencesFeb 1, 20240 citations

On pure MDS asymmetric entanglement-assisted quantum error-correcting codes

Ziteng Huang, Weijun Fang, Fang‐Wei Fu

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.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.

arXiv:2107.04166Jul 9, 20210 citations

d_{z} \geq d_{x}\geq 2

for all possible values of length

n

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,d_{z}/d_{x}]]_{q}

. For

q=\{2,3,4,5,7,8,9\}

, lists of optimal asymmetric CSS-like codes for reasonable lengths are presented.