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Journal of Physics Conference Series

Constructions of New Entanglement-Assisted Quantum Codes

Jianfa Qian, CLina Zhang

Published Jun 1, 2019• 0 citations
Publisher

Abstract

Abstract Entanglement-assisted quantum error-correcting codes are considered as a new research direction of quantum coding theory. In this work, based on cyclic codes, we present the constructions of new entanglement-assisted quantum error-correcting codes. Several new families of nonbinary entanglement-assisted quantum error-correcting codes with length n=2(q-1) can be efficiently constructed.

DOI
10.1088/1742-6596/1213/4/042007

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Entanglement-assisted quantum error-correcting codes, which can be seen as a generalization of quantum error-correcting codes, can be constructed from arbitrary classical linear codes by relaxing the self-orthogonality properties and using pre-shared entangled states between the sender and the receiver, and can also improve the performance of quantum error-correcting codes. In this paper, we construct some families of entanglement-assisted quantum maximum-distance-separable codes with parameters [[q2−1a,q2−1a−2d+2+c,d;c]]q[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q[[aq2−1​,aq2−1​−2d+2+c,d;c]]q​, where qqq is a prime power with the form q=am±ℓq = am \pm \ellq=am±ℓ, a=ℓ2−13a = \frac{{\ell^2} - 1}{3}a=3ℓ2−1​ is an odd integer, ℓ≡2 ( mod  6)\ell \equiv 2\ (\bmod\ 6)ℓ≡2 (mod 6) or ℓ≡4 ( mod  6)\ell \equiv 4\ (\bmod\ 6)ℓ≡4 (mod 6), and mmm is a positive integer. Most of these codes are new in the sense that their parameters are not covered by the codes available in the literature.
Entanglement-assisted quantum-error-correcting (EAQEC) codes are quantum codes which use entanglement as a resource. These codes can provide better error correction than the (entanglement unassisted) codes derived from the traditional stabilizer formalism. In this paper, we provide a general method to construct EAQEC codes from cyclic codes. Afterwards, the method is applied to Reed-Solomon codes, BCH codes, and general cyclic codes. We use the Euclidean and Hermitian construction of EAQEC codes. Three families have been created: two families of EAQEC codes are maximal distance separable (MDS), and one is almost MDS or almost near MDS. The comparison of the codes in this paper is mostly based on the quantum Singleton bound.
Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. Recently, constructing entanglement-assisted quantum maximum distance separable (EAQMDS) codes with flexible parameters has received much attention. In this work, four families of EAQMDS codes with a more general length are presented. And the method of selecting defining set is different from others. Compared with all the previously known results, the EAQMDS codes we constructed have larger minimum distance. All of these EAQMDS codes are new in the sense that their parameters are not covered by the quantum codes available in the literature.