Constructions of good entanglement-assisted quantum error correcting 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.

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 $n=\frac {q^{4m}-1}{q^{2}-1}$ .

The theory of entanglement-assisted quantum error-correcting codes (EAQECCs) is a generalization of the standard stabilizer formalism. Any quaternary (or binary) linear code can be used to construct EAQECCs under the entanglement-assisted (EA) formalism. We derive an EA-Griesmer bound for linear EAQECCs, which is a quantum analog of the Griesmer bound for classical codes. This EA-Griesmer bound is tighter than known bounds for EAQECCs in the literature. For a given quaternary linear code $$\mathcal {C}$$C, we show that the parameters of the EAQECC that EA-stabilized by the dual of $$\mathcal {C}$$C can be determined by a zero radical quaternary code induced from $$\mathcal {C}$$C, and a necessary condition under which a linear EAQECC may achieve the EA-Griesmer bound is also presented. We construct four families of optimal EAQECCs and then show the necessary condition for existence of EAQECCs is also sufficient for some low-dimensional linear EAQECCs. The four families of optimal EAQECCs are degenerate codes and go beyond earlier constructions. What is more, except four codes, our $$[[n,k,d_{ea};c]]$$[[n,k,dea?c]] codes are not equivalent to any $$[[n+c,k,d]]$$[[n+c,k,d]] standard QECCs and have better error-correcting ability than any $$[[n+c,k,d]]$$[[n+c,k,d]] QECCs.

The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using a decomposition of the 126-cap in PG(5,4), we firstly constructed LCD n-cap with 6≤n≤120. From these LCD caps, we then derived the related maximal entanglement [[n, n-6,4;6]] EAQECCs. Finally, using LCD LCD subcaps of 126-cap obtained, we constructed maximal entanglement EAQECCs with parameters [[n, n-;k,4;k]] for 6≤k≤11.

We develop the theory of entanglement-assisted quantum error correcting codes (EAQECCs), a generalization of the stabilizer formalism to the setting in which the sender and receiver have access to pre-shared entanglement. Conventional stabilizer codes are equivalent to self-orthogonal symplectic codes. In contrast, EAQECCs do not require self-orthogonality, which greatly simplifies their construction. We show how any classical quaternary block code can be made into a EAQECC. Furthermore, the error-correcting power of the quantum codes follows directly from the power of the classical codes.

The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform classical linear quaternary codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entangled qubits between the sender and the receiver. In this work, we give elementary recursive constructions of special quaternary codes of length n and dual distance four that constructed from known caps in projective space PG(5,4) and PG(6,4) for all length 6n283. Consequently, good maximal entanglement EAQECCs of minimum distance four for such length n are constructed from the obtained quaternary codes. Index Terms - EAQECCs, maximal entanglement, quaternary code, cap.

Entanglement-assisted quantum error-correcting (EAQEC) codes can be obtained from arbitrary classical linear codes, based on the entanglement-assisted stabilizer formalism. However, how to determine the required number of shared pairs is challenging. In this paper, we first construct three classes of classical linear MDS codes over finite fields by considering generalized Reed–Solomon codes and calculate the dimension of their Hermitian hulls. By using these MDS codes, we then obtain three new classes of EAQEC codes and EAQEC MDS codes, whose maximally entangled states can take various values. Moreover, these EAQEC codes have more flexible lengths.

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 (EAQECCs) make use of preexisting entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC 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 required ebits, which is an important basic parameter of an EAQECC, is usually calculated by computer search. In this paper, we construct four classes of MDS entanglement-assisted quantum error-correcting codes (MDS EAQECCs) based on k-Galois LCD MDS codes for some certain code lengths, where the parameter of ebits c can be easily computed algebraically and not by numerical search. Moreover, the constructed four classes of EAQECCs are also maximal-entanglement EAQECCs.

We give an infinite family of degenerate entanglement-assisted quantum error-correcting codes (EAQECCs) which violate the EA-quantum Hamming bound for non-degenerate EAQECCs and achieve the EA-quantum Singleton bound, thereby proving that the EA-quantum Hamming bound does not asymptotically hold for degenerate EAQECCs. Unlike the previously known quantum error-correcting codes that violate the quantum Hamming bound by exploiting maximally entangled pairs of qubits, our codes do not require local unitary operations on the entangled auxiliary qubits during encoding. The degenerate EAQECCs we present are constructed from classical error-correcting codes with poor minimum distances, which implies that, unlike the majority of known EAQECCs with large minimum distances, our EAQECCs take more advantage of degeneracy and rely less on the error correction capabilities of classical codes.

Entanglement-assisted quantum error-correcting codes (EAQECCs) make use of pre-existing entanglement between the sender and receiver to boost the rate of transmission. It is possible to construct an EAQECC from any classical linear code, unlike standard QECCs which can only be constructed from dual-containing codes. Operator quantum error-correcting codes (OQECCs) allow certain errors to be corrected (or prevented) passively, reducing the complexity of the correction procedure. We combine these two extensions of standard quantum error correction into a unified entanglement- assisted quantum error correction formalism. This new scheme, which we call entanglement-assisted operator quantum error correction (EAOQEC), is the most general and powerful quantum error-correcting technique known, retaining the advantages of both entanglement-assistance and passive correction. We present the formalism, show the considerable freedom in constructing EAOQECCs from classical codes, and demonstrate the construction with examples.

The entanglement-assisted (EA) formalism is a generalization of the standard stabilizer formalism, and it can transform arbitrary quaternary classical linear codes into entanglement-assisted quantum error correcting codes (EAQECCs) by using of shared entanglement between the sender and the receiver. Using the special structure of linear EAQECCs, we derive an EA-Plotkin bound for linear EAQECCs, which strengthens the previous known EA-Plotkin bound. This linear EA-Plotkin bound is tighter then the EA-Singleton bound, and matches the EA-Hamming bound and the EA-linear programming bound in some cases. We also construct three families of EAQECCs with good parameters. Some of these EAQECCs saturate this linear EA-Plotkin bound and the others are near optimal according to this bound; almost all of these linear EAQECCs are degenerate codes.

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 quantum MDS codes from constacyclic codes with large minimum distance

The dual of an entanglement-assisted quantum error-correcting (EAQEC) code is defined from the orthogonal group of a simplified stabilizer group. From the Poisson summation formula, this duality leads to the MacWilliams identities and linear programming bounds for EAQEC codes. We establish a table of upper and lower bounds on the minimum distance of any maximal-entanglement EAQEC code with length up to 15 channel qubits.