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IEEE Communications Letters

Optimal Entanglement-Assisted Quantum Codes With Larger Minimum Distance

Wan Jiang, Shixin Zhu, Xiaojing Chen

Published Sep 21, 2020• 5 citations
Publisher

Abstract

Entanglement-assisted quantum error-correcting code (EAQECC) is a new-type quantum code and can be constructed from arbitrary linear codes by sharing entangled states c in advance. In this letter, we explore a family of entanglement-assisted quantum error-correcting codes (EAQECCs) with length (q2- 1)/12 from negacyclic codes. Compared with known results in the literatures, our EAQECCs are new and have larger minimum distance.

DOI
10.1109/lcomm.2020.3025012

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Entanglement-assisted quantum error-correcting code (EAQECC) is a new-type quantum code and can be constructed from arbitrary linear codes by sharing entangled states c in advance. In this letter, we explore a family of entanglement-assisted quantum error-correcting codes (EAQECCs) with length (q2- 1)/12 from negacyclic codes. Compared with known results in the literatures, our EAQECCs are new and have larger minimum distance.
Entanglement-assisted quantum error-correcting codes (EAQECCs) can be obtained from arbitrary classical linear codes based on the entanglement-assisted stabilizer formalism, which greatly promoted the development of quantum coding theory. In this paper, we construct several families of [Formula: see text]-ary entanglement-assisted quantum maximum-distance-separable (EAQMDS) codes of lengths [Formula: see text] with flexible parameters as to the minimum distance [Formula: see text] and the number [Formula: see text] of maximally entangled states. Most of the obtained EAQMDS codes have larger minimum distances than the codes available in the literature.
The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer quantum error-correcting codes, which can be possibly constructed from any classical codes by relaxing the duality condition and utilizing preshared entanglement between the sender and receiver. In this paper, a new family of EAQECCs is constructed from negacyclic codes of length n=q2+1an=\frac{q^2+1}{a}n=aq2+1​, where qqq is an odd prime power, a=m2+12a=\frac{m^2+1}{2}a=2m2+1​ and mmm is an odd integer. Some new entanglement-assisted quantum maximum distance separable (EAQMDS) codes are obtained in the sense that their parameters are not covered by the previously known ones.
When constructing quantum codes under the entanglement-assisted (EA) stabilizer formalism, one can ignore the limitation of dual-containing condition. This allows us to construct EA quantum error-correcting codes (QECCs) from any classical linear codes. The main contribution of this manuscript is to make a general statement for determining the optimal number of pre-shared qubits instead of presenting only specific cases. Let [Formula: see text] and [Formula: see text], where [Formula: see text] is an odd prime power, [Formula: see text] and [Formula: see text]. By deeply investigating the decomposition of the defining set of negacyclic codes, we generalize the number of pre-shared entanglement pairs of Construction (1) in Lu et al. [Quantom Inf. Process. 17 (2018) 69] from [Formula: see text] to arbitrary even numbers less than or equal to [Formula: see text]. Consequently, a series of EA quantum maximum distance separable (EAQMDS) codes can be produced. The absolute majority of them are new and the minimum distance can be up to [Formula: see text]. Moreover, this method can be applied to construct many other families of EAQECCs with good parameters, especially large minimum distance.
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.
he entanglement-assisted stabilizer formalism provides a useful framework for constructing quantum error-correcting codes (QECC), which can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this paper, we construct five classes of entanglement-assisted quantum MDS (EAQMDS) codes based on classical MDS codes by exploiting one or more pre-shared maximally entangled states. We show that these EAQMDS codes have much larger minimum distance than the standard quantum MDS (QMDS) codes of the same length, and three classes of these EAQMDS codes consume only one pair of maximally entangled states.
Entanglement-assisted concatenated quantum codes (EACQCs) are constructed by concatenating two entanglement-assisted quantum error-correcting codes (EAQECCs). By selecting the inner and outer component codes carefully, it is able to construct state-of-the-art EACQCs with parameters better than previous quantum codes. In this work, we use almost maximum-distance-separable (MDS) codes and ℏ\hbarℏ-MDS codes as the outer codes to construct EACQCs. Because the range of code length of almost MDS and ℏ\hbarℏ-MDS codes is much more free than that of the commonly used MDS codes. We derive several families of new EACQCs with parameters better than the previously best known EAQECCs and standard quantum error-correcting codes (QECCs) of the same length and net transmissions. Moreover, we demonstrate that EACQCs are with maximal entanglement if both the inner and outer component codes are with maximal entanglement. As a result, we construct three new maximal-entanglement EACQCs which have optimal parameters. In addition, we present several new maximal-entanglement EACQCs whose minimum distance is only one less than the minimum distance of the optimal codes. In particular, we propose two new families of asymptotically good maximal-entanglement EACQCs with explicit constructions by using entanglement-assisted quantum algebraic geometry codes as the outer codes. At last, we prove that EACQCs can attain the quantum Gilbert-Varshamov bound for EAQECCs asymptotically.