Entanglement-assisted Quantum Error-correcting Codes Research Articles

Construction of some new entanglement-assisted quantum error-correcting codes of large lengths

The study of entanglement-assisted quantum error-correcting codes (EAQECCs) is currently a hot topic in coding theory and quantum information theory. As far as we know, there are few constructions of q-ary EAQECCs of large lengths, and most of the q-ary EAQECCs constructed in the literature have lengths less than or equal to q2+1. As a result, it will be interesting to construct q-ary EAQECCs of large lengths. The goal of this paper is to construct some good and new q-ary EAQECCs of large lengths from q2-ary matrix-product codes associated with the so-called non-singular by columns (NSC) quasi-unitary matrices. To this end, we first propose an effective method and a corresponding algorithm for constructing NSC quasi-unitary matrices. We then present an effective method for constructing EAQECCs of large lengths from matrix-product codes associated with NSC quasi-unitary matrices. Subsequently, we give the explicit construction of q-ary EAQECCs of lengths kn with n≤q2+1, k=k0 for q≥k0, where k0=3,4,5,7,8,9,11,13,16,17,19, and k=q+2, q2−1, q2 for 3≤q≤19. Finally, some new q-ary EAQECCs of large lengths such as lengths 3(q2−1), 3q2, 4(q2−1), 4q2, q3, q(q2+1), q2(q+2), (q2+1)(q+2), q2(q2−1), q4−1, q4, q2(q2+1) for some q are constructed, which are not covered by previously known ones.

On the ℓ-DLIPs of codes over finite commutative rings

Generalizing the linear complementary duals, the linear complementary pairs and the hull of codes, we introduce the concept of ℓ-dimension linear intersection pairs (ℓ-DLIPs) of codes over a finite commutative ring (R), for some positive integer ℓ. In this paper, we study ℓ-DLIP of codes over R in a very general setting by a uniform method. Besides, we provide a necessary and sufficient condition for the existence of a non-free (or free) ℓ-DLIP of codes over a finite commutative Frobenius ring. In addition, we obtain a generator set of the intersection of two constacyclic codes over a finite chain ring which helps us to get an important characterization of ℓ-DLIP of constacyclic codes. Finally, the ℓ-DLIP of constacyclic codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting (EAQEC) codes.

Entanglement-assisted quantum error-correcting codes from subfield subcodes of projective Reed–Solomon codes

We study the subfield subcodes of projective Reed–Solomon codes and their duals: we provide bases for these codes and estimate their parameters. With this knowledge, we can construct symmetric and asymmetric entanglement-assisted quantum error-correcting codes, which in many cases have new or better parameters than the ones available in the literature.

On the hulls of cyclic codes of oddly even length over [formula omitted

In this paper, we study hulls of cyclic codes of length 2n (i.e., oddly even length) over the ring Z4 of integers modulo 4 by viewing these codes as ideals of the quotient ring Z4[x]/<x2n−1>, where n is an odd positive integer. We express the generator polynomials of the hull of each cyclic code C of length 2n over Z4 in terms of the generator polynomials of the code C, and we note that the 2-dimension of the hull of the code C is an integer Θ satisfying 0≤Θ≤2n. We also obtain an enumeration formula for cyclic codes of length 2n over Z4 with hulls of a given 2-dimension. Besides this, we study the average 2-dimension, denoted by E(2n), of the hulls of cyclic codes of length 2n over Z4 and establish a formula for E(2n) that handles well. With the help of this formula, we deduce that E(2n)=5n6 when n∈N2, and we study the growth rate of E(2n) with respect to the length 2n when n∉N2, where N2 denotes the set of all positive integers ω such that w divides 2i+1 for some positive integer i. Further, when n∉N2, we derive lower and upper bounds on E(2n) and show that these bounds are attained at n=7. We also illustrate these results with some examples. As an application of these results, we construct some entanglement-assisted quantum error-correcting codes (EAQECCs) over Z4 with specific parameters from cyclic codes of oddly even length over Z4.

Linear codes of larger lengths with Galois hulls of arbitrary dimensions and related entanglement-assisted quantum error-correcting codes

Galois hulls are generalizations of Euclidean and Hermitian hulls, which have attracted interest because of their important applications in determining the complexity of some algorithms about linear codes and in constructing entanglement-assisted quantum error-correcting codes (EAQECCs). In this paper, the whole contribution is two-folded. On one hand, we propose explicit methods to construct linear codes of larger lengths with Galois hulls of arbitrary dimensions from given Galois self-orthogonal codes. Conditions required in our approach are proven to be relatively weak. On the other hand, we apply these results to construct EAQECCs. Two bounds for EAQECCs constructed from linear codes with prescribed dimensional Galois hull are given and EAQECCs with rates greater than or equal to 12 and positive net rates can be obtained. We also present many interesting examples to explain visually how these two aspects work.

Steane enlargement of entanglement-assisted quantum error-correcting codes

We introduce a Steane-like enlargement procedure for entanglement-assisted quantum error-correcting codes (EAQECCs) obtained by considering Euclidean inner product. We give formulae for the parameters of these enlarged codes and apply our results to explicitly compute the parameters of enlarged EAQECCs coming from some BCH codes.

Transforming collections of Pauli operators into equivalent collections of Pauli operators over minimal registers

Transformations which convert between fermionic modes and qubit operations have become a ubiquitous tool in quantum algorithms for simulating systems. Similarly, collections of Pauli operators might be obtained from solutions of nonlocal games and satisfiability problems. Drawing on ideas from entanglement-assisted quantum error-correcting codes and quantum convolutional codes, we prove the obtainable lower bound for the number of qubits needed to represent such Pauli operations which are equivalent to the initial ones and provide a procedure for determining such a set of minimal register Pauli operations.

New MDS EAQECCs derived from constacyclic codes over [formula omitted

Entanglement-assisted quantum error-correcting codes (EAQECCs) can improve information rate by using pre-shared entanglement between the sender and the receiver. In this paper, we construct three new infinite families of maximum distance separable (MDS) EAQECCs with less pre-shard maximally entangled states based on the Gray images of constacyclic codes over the finite non-chain ring Fq2+vFq2 with v2=v. The parameters of these MDS EAQECCs are not covered by the codes available in the previous literatures. Furthermore, our results generalize and improve the work of Lu et al. (2018) [16], (2020) [19].

New entanglement-assisted quantum MDS codes derived from cyclic codes

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 $[[\frac{{{q^2} - 1}}{a},\frac{{{q^2} - 1}}{a} - 2d+2 + c,d;c]]_q$, where $q$ is a prime power with the form $q = am \pm \ell$, $a = \frac{{\ell^2} - 1}{3}$ is an odd integer, $\ell \equiv 2\ (\bmod\ 6)$ or $\ell \equiv 4\ (\bmod\ 6)$, and $m$ 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.

Skew Constacyclic Codes over a Non-Chain Ring

In this paper, we investigate the algebraic structure of the non-local ring Rq=Fq[v]/⟨v2+1⟩ and identify the automorphisms of this ring to study the algebraic structure of the skew constacyclic codes and their duals over this ring. Furthermore, we give a necessary and sufficient condition for the skew constacyclic codes over Rq to be linear complementary dual (LCD). We present some examples of Euclidean LCD codes over Rq and tabulate the parameters of Euclidean LCD codes over finite fields as the Φ-images of these codes over Rq, which are almost maximum distance separable (MDS) and near MDS. Eventually, by making use of Hermitian linear complementary duals of skew constacyclic codes over Rq and the map Φ, we give a class of entanglement-assisted quantum error correcting codes (EAQECCs) with maximal entanglement and tabulate parameters of some EAQECCs with maximal entanglement over finite fields.

Hulls of double cyclic codes

In this paper, we characterize hulls of double cyclic codes over Z2. Firstly, we give the generators of hulls of double cyclic codes under the separable case and the non-separable case with coprime odd block lengths. Then, using the concept of a generating function in combinatorics, the solution of dimensions of hulls of double cyclic codes over Z2 is given. Furthermore, the enumeration of double cyclic codes over Z2 with hulls of a fixed dimension is determined. As an application, some good entanglement-assisted quantum error-correcting codes (EAQECCs) are obtained by the hulls of non-separable double cyclic codes.

Galois hulls of special Goppa codes and related codes with application to EAQECCs

Galois hulls of MDS codes can be applied to construst MDS entanglement-assisted quantum error-correcting codes (EAQECCs). Goppa codes and expurgated Goppa codes (resp., extended Goppa codes) over Fqm are GRS codes (resp., extended GRS codes) when m=1. In this paper, we investigate the Galois dual codes of a special kind of Goppa codes and related codes and provide a necessary and sufficient condition for the Galois dual codes of such codes to be Goppa codes and related codes. Then we determine the Galois hulls of the above codes. In particular, we completely characterize Galois LCD, Galois self-orthogonal, Galois dual-containing and Galois self-dual codes among such family of codes. Moreover, we apply the above results to EAQECCs.

New entanglement-assisted quantum codes constructed from Hermitian LCD codes

&lt;abstract&gt;&lt;p&gt;Hermitian linear complementary dual (LCD) codes are a class of linear codes that intersect with their Hermitian dual trivially. Each Hermitian LCD code can give an entanglement-assisted quantum error-correcting code (EAQECC) with maximal entanglement. Methods of constructing Hermitian LCD codes from known codes were developed, and seven new Hermitian LCD codes with parameters $ [119,4,88]_{4} $, $ [123,4,91]_{4} $, $ [124,4,92]_{4} $, $ [136,4,101]_{4} $, $ [140,4,104]_{4} $, $ [188,4,140]_{4} $ and $ [212,4,158]_{4} $ were constructed. Seven families of Hermitian LCD codes and their related EAQECCs were derived from these codes. These new EAQECCs have better parameters than those known in the literature.&lt;/p&gt;&lt;/abstract&gt;

An algorithmic approach to entanglement-assisted quantum error-correcting codes from the Hermitian curve

&lt;p style='text-indent:20px;'&gt;We study entanglement-assisted quantum error-correcting codes (EAQECCs) arising from classical one-point algebraic geometry codes from the Hermitian curve with respect to the Hermitian inner product. Their only unknown parameter is &lt;inline-formula&gt;&lt;tex-math id="M1"&gt;\begin{document}$ c $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt;, the number of required maximally entangled quantum states since the Hermitian dual of an AG code is unknown. In this article, we present an efficient algorithmic approach for computing &lt;inline-formula&gt;&lt;tex-math id="M2"&gt;\begin{document}$ c $\end{document}&lt;/tex-math&gt;&lt;/inline-formula&gt; for this family of EAQECCs. As a result, this algorithm allows us to provide EAQECCs with excellent parameters over any field size.&lt;/p&gt;

Multivariate Goppa Codes

In this paper, we introduce multivariate Goppa codes, which contain, as a particular case, the well-known classical Goppa codes. We provide a parity check matrix for a multivariate Goppa code in terms of a tensor product of generalized Reed-Solomon codes. We prove that multivariate Goppa codes are subfield subcodes of augmented Cartesian codes. By showing how this new family of codes relates to a tensor product of generalized Reed-Solomon codes and augmented codes, we obtain information about the parameters, subcodes, duals, and hulls of multivariate Goppa codes. We see that in some instances, the hulls of multivariate Goppa codes (resp., tensor product of generalized Reed-Solomon codes) are also multivariate Goppa codes (resp. tensor product of generalized Reed-Solomon codes). We utilize the multivariate Goppa codes to obtain entanglement-assisted quantum error-correcting codes and to build families of long LCD, self-dual, or self-orthogonal codes.

Entanglement-Assisted Quantum Codes from Cyclic Codes

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.

On cyclic and negacyclic codes with one-dimensional hulls and their applications

Linear codes over finite fields with small dimensional hulls have received much attention due to their applications in cryptology and quantum computing. In this paper, we study cyclic and negacyclic codes with one-dimensional hulls. We determine precisely when cyclic and negacyclic codes over finite fields with one-dimensional hulls exist. We also introduce one-dimensional linear complementary pairs of cyclic and negacyclic codes. As an application, we obtain numerous optimal or near optimal cyclic codes with one-dimensional hulls over different fields and, by using these codes, we present new entanglement-assisted quantum error-correcting codes (EAQECCs). In particular, some of these EAQEC codes are maximal distance separable (MDS). We also obtain one-dimensional linear complementary pairs of cyclic codes, which are either optimal or near optimal.

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Quantum error-correcting codes play the role of suppressing noise and decoherence in quantum systems by introducing redundancy. Some strategies can be used to improve the parameters of these codes. For example, entanglement can provide a way for quantum error-correcting codes to achieve higher rates than the one obtained by means of the traditional stabilizer formalism. Such codes are called entanglement-assisted quantum error-correcting (EAQEC) codes. In this paper, we utilize algebraic geometry codes to construct several families of EAQEC codes derived from the Euclidean and the Hermitian construction. Three families constructed here consist of codes whose quantum Singleton defect is equal to zero, one, or two. We also construct families of EAQEC codes with an encoding rate exceeding the quantum Gilbert-Varshamov bound. Additionally, asymptotically good towers of linear complementary dual codes are used to obtain asymptotically good families of EAQEC codes consuming maximal entanglement. Furthermore, a simple comparison with the quantum Gilbert-Varshamov bound demonstrates that, by utilizing the proposed construction, it is possible to generate an asymptotically family of EAQEC codes that exceeds this bound.

Cyclic codes and some new entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error correcting codes (EAQECCs) play a significant role in protecting quantum information from decoherence and quantum noise. In this work, we construct six families of new EAQECCs of lengths $$n=(q^2+1)/a$$ , $$n=q^2+1$$ and $$n=(q^2+1)/2$$ from cyclic codes, where $$a=m^2+1$$ ( $$m\ge 1$$ is odd) and q is an odd prime power with the form of $$a|(q+m)$$ or $$a|(q-m)$$ . Moreover, those EAQECCs are entanglement-assisted quantum maximum distance separable (EAQMDS) codes when $$d\le (n+2)/2$$ . In particular, the length of EAQECCs we studied is more general and the method of selecting defining set is different from others. Compared with all the previously known results, the EAQECCs in this work have flexible parameters and larger minimum distance. All of these EAQECCs are new in the sense that their parameters are not covered by the quantum codes available in the literature.

New Galois Hulls of GRS Codes and Application to EAQECCs

Galois hulls of linear codes have important applications in quantum coding theory. In this paper, we construct some new classes of (extended) generalized Reed-Solomon (GRS) codes with Galois hulls of arbitrary dimensions. We also propose a general method on constructing GRS codes with Galois hulls of arbitrary dimensions from special Euclidean orthogonal GRS codes. Finally, we construct several new families of entanglement-assisted quantum error-correcting codes (EAQECCs) and MDS EAQECCs by utilizing the above results.