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arXiv:2107.04166

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

Ziteng Huang, Weijun Fang, Fang‐Wei Fu

Published Jul 9, 2021• 0 citations
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Abstract

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.

DOI
10.48550/arxiv.2107.04166

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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.
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.
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.
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.
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.
Entanglement-assisted quantum error-correcting (EAQEC) codes make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQEC code from any classical linear code, unlike standard quantum error-correcting codes, which can only be constructed from dual-containing codes. However, the number ccc of pre-shared maximally entangled states is usually calculated by computer search. In this paper, we first give a new formula for calculating the number ccc of pre-shared maximally entangled states. Then, using this formula, we construct three classes of new entanglement-assisted quantum error-correcting maximum-distance-separable ( EAQEC MDS) codes.