Entanglement-assisted Quantum Error Research Articles

Construction of Hermitian Self-Orthogonal Codes and Application

We introduce some methods for constructing quaternary Hermitian self-orthogonal (HSO) codes, and construct quaternary [n, 5] HSO for 342≤n≤492. Furthermore, we present methods of constructing Hermitian linear complementary dual (HLCD) codes from known HSO codes, and obtain many HLCD codes with good parameters. As an application, 31 classes of entanglement-assisted quantum error correction codes (EAQECCs) with maximal entanglement can be obtained from these HLCD codes. These new EAQECCs have better parameters than those in the literature.

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.

Performance of entanglement-assisted quantum codes with noisy ebits over asymmetric and memory channels

Entanglement-assisted quantum error correction codes (EAQECCs) play an important role in quantum communications with noise. Such a scheme can use arbitrary classical linear code to transmit qubits over noisy quantum channels by consuming some ebits between the sender (Alice) and the receiver (Bob). It is usually assumed that the preshared ebits of Bob are error free. However, noise on these ebits is unavoidable in many cases. In this work, we evaluate the performance of EAQECCs with noisy ebits over asymmetric quantum channels and quantum channels with memory by computing the exact entanglement fidelity of several EAQECCs. We consider asymmetric errors in both qubits and ebits and show that the performance of EAQECCs in entanglement fidelity gets improved for qubits and ebits over asymmetric channels. In quantum memory channels, we compute the entanglement fidelity of several EAQECCs over Markovian quantum memory channels and show that the performance of EAQECCs is lowered down by the channel memory. Furthermore, we show that the performance of EAQECCs is diverse when the error probabilities of qubits and ebits are different. In both asymmetric and memory quantum channels, we show that the performance of EAQECCs is improved largely when the error probability of ebits is reasonably smaller than that of qubits.

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.

L-LIPs of codes over finite chain rings

In this paper, we first generalize the l-linear intersection pair (l-LIP) of codes over finite fields to the linear l-intersection pair of codes over finite chain rings. Then three criteria of l-linear intersection pair of codes over finite chain rings are obtained. By means of the generator set of cyclic codes, we obtain the generator set of the intersection of two cyclic codes which helps us to get a characterization of l-cyclic intersection pair of codes. As an application, l-cyclic intersection pair of codes over finite chain rings are used to construct new entanglement-assisted quantum error correcting codes. Finally, under suitable conditions, we also obtain a characterization for a l-cyclic intersection pair of codes over principal ideal rings Zk.

MDS Codes With Galois Hulls of Arbitrary Dimensions and the Related Entanglement-Assisted Quantum Error Correction

Let <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q=p^{e}$ </tex-math></inline-formula> be a prime power and <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> be an integer with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$0\leq \ell \leq e-1$ </tex-math></inline-formula> . The <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -Galois hull of classical linear codes is a generalization of the Euclidean hull and Hermitian hull. We provide a necessary and sufficient condition under which a codeword of a GRS code or an extended GRS code belongs to its <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -Galois dual code, generalizing both the Euclidean case and Hermitian case in the literature. By using four different tools: 1) the norm mapping from <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}^{\ast }$ </tex-math></inline-formula> to <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{p^{\ell }}^{\ast }$ </tex-math></inline-formula> ; 2) the direct product of two cyclic subgroups; 3) the coset decomposition of a cyclic group; 4) an additive subgroup of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\mathbb {F}_{q}$ </tex-math></inline-formula> and its cosets, we construct eleven families of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary MDS codes with <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -Galois hulls of arbitrary dimensions, and give the related eleven families of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$[[n,k,d;c]]_{q}$ </tex-math></inline-formula> entanglement-assisted quantum error-correcting codes (EAQECCs) with relatively large minimum distance in the sense that <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2d=n-k+2+c$ </tex-math></inline-formula> . We show that developing the theory on <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> -Galois hulls of <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary MDS codes in this paper enables us to obtain new <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$q$ </tex-math></inline-formula> -ary EAQECCs with different kinds of length sets via different <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$\ell $ </tex-math></inline-formula> , where <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"> <tex-math notation="LaTeX">$2\ell \mid e$ </tex-math></inline-formula> .

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.

Entanglement-assisted quantum error-correcting codes from RS codes and BCH codes with extension degree 2

Entanglement-assisted quantum error correcting codes (EAQECCs) constructed from Reed-Solomon codes and BCH codes are considered in this work. It is provided a complete and explicit formula for the parameters of EAQECCs coming from any Reed-Solomon code, for the Hermitian metric, and from any BCH code with extension degree $2$ and consecutive cyclotomic cosets, for both the Euclidean and the Hermitian metric. The main task in this work is the computation of a completely general formula for $c$, the minimum number of required maximally entangled quantum states.

Hulls of Generalized Reed-Solomon Codes via Goppa Codes and Their Applications to Quantum Codes

A Goppa code over \Bbb F <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">q^m</sub> is a well-known subclass of algebraic error-correcting code. If m=1, then it is a generalized Reed-Solomon(GRS) code and its dual code is called a GRS code via a Goppa code. In this paper, we give a necessary and sufficient condition that the dual codes of GRS codes via (expurgated) Goppa codes are also GRS codes via Goppa codes. Under the above condition, we show that the hulls of GRS codes via Goppa codes are still GRS codes via Goppa codes. As an application, we characterize LCD GRS codes and self-dual GRS codes under the above condition. Some numerical examples are also presented to illustrate our main results. Moreover, we also apply our result to entanglement-assisted quantum error correcting codes (EAQECCs) and obtain two new families of MDS EAQECCs with arbitrary parameters.

Construction of entanglement-assisted quantum MDS codes

Entanglement-assisted quantum error correction codes (EAQEECs) can improve the reliability of transmission, which make use of pre-existing entanglement between two parties. In this paper, we construct two classes of entanglement-assisted quantum maximum distance separable (EAQMDS) codes via generalized Reed-Solomon (GRS) codes. Consequently, the results show that some of these EAQMDS codes have much larger minimum distance than ones available in the literature. Meanwhile, many of these EAQMDS codes are new in the sense that the parameters of these codes are not covered by the previously known ones. ()

New entanglement-assisted quantum MDS codes with length $$n=\frac{q^2+1}{5}$$

The entanglement-assisted stabilizer formalism can transform arbitrary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs). In this work, we construct some new entanglement-assisted quantum MDS (EAQMDS) codes with length $n=\frac{q^2+1}5$ from cyclic codes. Compared with all the previously known parameters with the same length, all of them have flexible parameters and larger minimum distance.

Two Families of Entanglement-Assisted Quantum MDS Codes from Constacyclic Codes

Entanglement-assisted quantum error correcting codes (EAQECCs) can be derived from arbitrary classical linear codes. However, it is a very difficult task to determine the number of entangled states required. In this work, using the method of the decomposition of the defining set of constacyclic codes, we construct two families 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 q. 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.

Linear $$\ell $$-intersection pairs of codes and their applications

In this paper, a linear $$\ell $$-intersection pair of codes is introduced as a generalization of linear complementary pairs of codes. Two linear codes are said to be a linear $$\ell $$-intersection pair if their intersection has dimension $$\ell $$. Characterizations and constructions of such pairs of codes are given in terms of the corresponding generator and parity-check matrices. Linear $$\ell $$-intersection pairs of MDS codes over $${\mathbb {F}}_q$$ of length up to $$q+1$$ are given for all possible parameters. As an application, linear $$\ell $$-intersection pairs of codes are used to construct entanglement-assisted quantum error correcting codes. This provides a large number of new MDS entanglement-assisted quantum error correcting codes.

Entanglement-assisted quantum error correction codes with length $$n=q^2+1$$

In this paper, by investigating $$q^2$$ -cyclotomic coset modulo rn in detail, where q is a prime power, $$n=q^2+1$$ and $$r\mid (q+1)$$ , series of entanglement-assisted quantum error correction (EAQEC) codes with flexible parameters of length n are constructed from constacyclic codes (including cyclic codes). Most of our EAQEC codes are new and have large minimum distance. As to EAQEC codes constructed from cyclic codes, their all possible parameters are determined completely. When minimum distance $$d\le \frac{n+2}{2}$$ , all of our constructed EAQEC codes are entanglement-assisted quantum MDS (EAQMDS) codes. Those previously known EAQMDS codes with the same length in Fan et al. (Quantum Inf Comput 16:423–434, 2016), Chen et al. (Quantum Inf Process 16(303):1–22, 2017), Lu et al. (Finite Fields Their Appl 53:309–325, 2018), Mustafa and Emre (Comput Appl Math 38(75):1–13, 2019) and Qian and Zhang (Quantum Inf Process 18(71):1–12, 2019) are special cases of ours. Besides, some maximum entanglement EAQEC codes and maximum entanglement EAQMDS codes are derived as well.

Entanglement-Assisted Quantum Negacyclic BCH Codes

The entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and important class of quantum codes. The entanglement-assisted formalism can transform arbitrary classical linear codes into EAQECCs by using pre-shared entanglement between the sender and the receiver. In this paper, by decomposing the defining set of negacyclic BCH codes, we construct a class of new EAQECCs with length $n=\frac {q^{4m}-1}{q^{2}-1}$ .

Entanglement-assisted quantum MDS codes from constacyclic codes with large minimum distance

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.

Minimal-Entanglement Entanglement-Assisted Quantum Error Correction Codes from Modified Circulant Matrices

In this paper, new construction methods of entanglement-assisted quantum error correction code (EAQECC) from circulant matrices are proposed. We first construct the matrices from two vectors of constraint size, and determine the isotropic subgroup. Then, we also propose a method for calculation of the entanglement subgroup based on standard forms of binary matrices to satisfy the constraint conditions of EAQECC. With isotropic and entanglement subgroups, we determine all the parameters and the minimum distance of the EAQECC. The proposed EAQECC with small lengths are presented to explain the practicality of this construction of EAQECC. Comparison with some earlier constructions of EAQECC shows that the proposed EAQECC is better.

Constructions of good entanglement-assisted quantum error correcting codes

Entanglement-assisted quantum error correcting codes (EAQECCs) are a simple and fundamental class of codes. They allow for the construction of quantum codes from classical codes by relaxing the duality condition and using pre-shared entanglement between the sender and receiver. However, in general it is not easy to determine the number of shared pairs required to construct an EAQECC. In this paper, we show that this number is related to the hull of the classical code. Using this fact, we give methods to construct EAQECCs requiring desirable amounts of entanglement. This allows for designing families of EAQECCs with good error performance. Moreover, we construct maximal entanglement EAQECCs from LCD codes. Finally, we prove the existence of asymptotically good EAQECCs in the odd characteristic case.

Constructions of q-ary entanglement-assisted quantum MDS codes with minimum distance greater than q+1

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.

Maximal entanglement entanglement-assisted quantum codes of distance three

Entanglement-assisted quantum error correcting code (EAQECC) is a generalization of standard stabilizer quantum code. Maximal entanglement EAQECCs can achieve the EA-hashing bound asymptotically. In this work, we give elementary recursive constructions of quaternary zero radical codes with dual distance three for all n ≥ 4. Consequently, good maximal entanglement EAQECCs of minimum distance three for such length n are obtained. Almost all of these EAQECCs are optimal or near optimal according to the EA-quantum Hamming bound.