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Asymmetric Entanglement-Assisted Quantum MDS Codes Constructed from Constacyclic Codes

Jian‐Zhang Chen, Wanchuan Fang, Shuo Zhou, J. F. Qiu, Chenyang Zhang, Yixin Xu, Bozhe Zeng, Youqin Chen

Published Nov 30, 2023• 10 citations
Publisher

Abstract

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.

DOI
10.3390/e25121603

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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 entanglement-assisted (EA) formalism allows arbitrary classical linear codes to transform into entanglement-assisted quantum error correcting codes (EAQECCs) by using pre-shared entanglement between the sender and the receiver. In this work, we propose a decomposition of the defining set of constacyclic codes. Using this method, we construct four classes of q-ary entanglement-assisted quantum MDS (EAQMDS) codes based on classical constacyclic MDS codes by exploiting less pre-shared maximally entangled states. We show that a class of q-ary EAQMDS have minimum distance upper bound greater than 3q−1. Some of them have much larger minimum distance than the known quantum MDS (QMDS) codes of the same length. Most of these q-ary EAQMDS codes are new in the sense that their parameters are not covered by the codes available in the literature.
Although quantum maximal-distance-separable (MDS) codes that satisfy the quantum singleton bound have become an important research topic in the quantum coding theory, it is not an easy task to search for quantum MDS codes with the minimum distance that is larger than (q/2) + 1. The pre-shared entanglement between the sender and the receiver can improve the minimum distance of quantum MDS codes such that the minimum distance of some constructed codes achieves (q/2) + 1 or exceeds (q/2) + 1. Meanwhile, how to determine the required number of maximally entangled states to make the minimum distance of quantum MDS codes larger than (q/2) + 1 is an interesting problem in the quantum coding theory. In this paper, we utilize the decomposition of the defining set and q2-cyclotomic cosets of constacyclic codes with the form q = αm + t or q = αm + α - t and n = (q2+ 1/α) to construct some new families of entanglement-assisted quantum MDS codes that satisfy the entanglement-assisted quantum singleton bound, where q is an odd prime power and m is a positive integer, while both α and t are positive integers such that α = t2+ 1. The parameters of these codes constructed in this paper are more general compared with the ones in the literature. Moreover, the minimum distance of some codes in this paper is larger than (q/2) + 1 or q + 1.
Generally, it is not easy to construct quantum maximal-distance-separable (MDS) codes with the minimum distance greater thanq2+1\frac {q}{2}+12q​+1. The minimum distance of quantum MDS codes can achieveq2+1\frac {q}{2}+12q​+1or exceedq2+1\frac {q}{2}+12q​+1by adopting pre-shared entanglement. In this work, some new families of entanglement-assisted quantum MDS codes that satisfy the quantum Singleton bound are constructed and the number of maximally entangled states required is determined to make the minimum distance of some constructed codes achieveq2+1\frac {q}{2}+12q​+1or exceedq2+1\frac {q}{2}+12q​+1by utilizing the decomposition of the defining set andq2q^{2}q2-cyclotomic cosets of constacyclic codes with lengthq2+1γ\frac {q^{2}+1}{\gamma }γq2+1​, whereγ=t2+1\gamma =t^{2}+1γ=t2+1,tttis a power of 2 andq=te>4q=t^{e}>4q=te>4withe≡1 mod 4e\equiv 1~\textrm {mod}~4e≡1 mod 4ore≡3 mod 4e\equiv 3~\textrm {mod}~4e≡3 mod 4. Moreover, the parameters of these codes constructed in this paper are more general relative to the ones in the literature and the minimum distance of some codes constructed in this paper is larger thanq2+1\frac {q}{2}+12q​+1.
Entanglement-assisted quantum error-correcting codes can be seen as a new-type of quantum error-correcting codes and can be constructed from arbitrary linear codes which should not satisfy the dual-containing condition by utilizing shared entangled states between the sender and the receiver in advance. In this paper, we construct several new classes of entanglement-assisted quantum maximum-distance-separable codes from constacyclic codes and cyclic codes by exploiting small pre-shared entangled states, respectively. These codes are new in the sense that they are not covered by the codes available in the literature.
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.
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.