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International Journal of Modern Physics C

Entanglement-assisted quantum error-correcting codes and asymmetric entanglement-assisted quantum error-correcting codes of length 5pt over 𝔽pm

Hai Quang Dinh, Bac Trong Nguyen, Van‐Nui Nguyen, Hiep L. Thi

Published Feb 27, 2026• 0 citations

Abstract

Classical cyclic and constacyclic codes play a fundamental role in coding theory, thanks to their rich algebraic structures and efficient encoding/decoding algorithms. These codes are especially important in the construction of quantum error-correcting codes, since their inherent algebraic properties facilitate the derivation of self-orthogonality or dual-containing conditions that are crucial for ensuring quantum error correction capability. In this paper, we construct entanglement-assisted quantum error-correcting codes (EAQEC codes) of length [Formula: see text] over [Formula: see text]. We compare our EAQEC codes with all known EAQEC codes to see that our EAQEC codes are new in the sense that their parameters are different from all the previous constructions. Moreover, our EAQEC codes have shorter lengths than the known codes, yet exhibit significantly larger quantum distances. This is highly meaningful in the context of quantum error correction, as it enables the construction of codes with small lengths while maintaining large quantum distances. We also construct asymmetric EAQEC (AEAQEC) codes of length [Formula: see text] over [Formula: see text]. We evaluate our AEAQEC codes against all existing ones and find that they are both new and better.

DOI: 10.1142/s0129183126430023

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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 (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

c

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

c

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.

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)

q

-qubit quantum codes attaining the quantum Singleton bound were constructed for very restricted lengths

n \leq q^2+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

codes for arbitrary

n

satisfying

n \leq q^2+1

and arbitrary distance

d\leq \frac{n+2}{2}

. It is proved that for any given length

n

satisfying

O(q^2)=n \leq q^2+1

and any given distance

d

satisfying $ O(q^2)=d \leq \frac{n+2}{2}$, there exist at least

O(q^2)

MDS EAQEC

[[n, k, d, c]]_q

codes with different

d

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.

We develop the theory of entanglement-assisted quantum error-correcting (EAQEC) codes, a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to preshared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQEC codes do not require self-orthogonality, which greatly simplifies their construction. We show how any classical binary or quaternary block code can be made into an EAQEC code. We provide a table of best known EAQEC codes with code length up to 10. With the self-orthogonality constraint removed, we see that the distance of an EAQEC code can be better than any standard quantum error-correcting code with the same fixed net yield. In a quantum computation setting, EAQEC codes give rise to catalytic quantum codes, which assume a subset of the qubits are noiseless. We also give an alternative construction of EAQEC codes by making classical entanglement-assisted codes coherent.

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.