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arXiv:2408.09334

EAQEC codes from s-Galois hulls decomposition of linear codes

Hui Li, Xiusheng Liu

Published Aug 18, 2024• 0 citations
arXivHTMLPDF

Abstract

Entanglement-assisted quantum error-correcting (EAQEC) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, we first give a s-Galois hulls decomposition of linear codes over finite fields. By means of this decomposition, we provide a method to construct EAQEC codes. Using this method, we construct new EAQEC codes.

DOI
10.48550/arxiv.2408.09334

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Entanglement-assisted quantum error-correcting (EAQEC) codes are a subclass of quantum error-correcting codes which use entanglement as a resource. These codes can provide error correction capability higher than the codes derived from the traditional stabilizer formalism. In this paper, we first give a s-Galois hulls decomposition of linear codes over finite fields. By means of this decomposition, we provide a method to construct EAQEC codes. Using this method, we construct new EAQEC codes.
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The Galois hull of a linear code is the intersection of itself and its Galois dual code, which has aroused the interest of researchers in these years. In this paper, we study Galois hulls of linear codes. Firstly, the symmetry of the dimensions of Galois hulls of linear codes is found. Some new necessary and sufficient conditions for linear codes being Galois self-orthogonal codes, Galois self-dual codes, and Galois linear complementary dual codes are characterized. Then, we propose explicit methods to construct Galois self-orthogonal codes of larger length from given Galois self-orthogonal codes. As an application, linear codes of larger length with Galois hulls of arbitrary dimensions are further derived. Focusing on the Hermitian inner product, two new classes of Hermitian self-orthogonal maximum distance separable (MDS) codes are also constructed. Finally, applying all the results to the construction of entanglement-assisted quantum error-correcting codes (EAQECCs), many new qqq-ary or q\sqrt{q}q​-ary EAQECCs and MDS EAQECCs with rates greater than or equal to 12\frac{1}{2}21​ and positive net rates can be obtained. Moreover, the minimum distance of many q\sqrt{q}q​-ary MDS EAQECCs of length n>q+1n>\sqrt{q}+1n>q​+1 is greater than or equal to ⌈q2⌉\lceil \frac{\sqrt{q}}{2} \rceil⌈2q​​⌉.
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.
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 hull of linear codes plays an important role in quantum information and coding theory. In the present paper, by investigating the Galois hulls of generalized Reed-Solomon (GRS) codes and extended GRS codes over the finite field Fq, we give several new families of MDS codes with Galois hulls of arbitrary dimensions that are not obtained before. Some of them generalize the ones in the literature [13]. As a consequence, using these MDS codes with Galois hulls of arbitrary dimensions, we construct nine new families of MDS entanglement-assisted quantum error-correcting codes (EAQECCs) with flexible parameters.
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 ccc 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 ccc 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.