arXiv:1612.07150

Good and asymptotically good quantum codes derived from algebraic geometry codes

Giuliano G. La Guardia, Francisco Revson F. Pereira

Published Dec 21, 2016• 2 citations

Abstract

In this paper we construct several new families of quantum codes with good and asymptotically good parameters. These new quantum codes are derived from (classical) algebraic geometry (AG) codes by applying the Calderbank-Shor-Steane (CSS) construction. Many of these codes have large minimum distances when compared with its code length and they have relatively small Singleton defect. For example, we construct a family [[46; 2(t_2 - t1); d]]_25 of quantum codes, where t_1, t_2 are positive integers such that 1 < t_1 < t_2 < 23 and d >= min {46 - 2t_2, 2t_1-2}, of length n = 46, with minimum distance in the range 2 =< d =< 20, having Singleton defect four. Additionally, by utilizing t-point AG codes, with t >= 2, we show how to obtain sequences of asymptotically good quantum codes.

DOI: 10.48550/arxiv.1612.07150

Related papers

Classical and Quantum Convolutional Codes Derived From Algebraic Geometry Codes

Francisco Revson F. Pereira, Giuliano G. La Guardia, Francisco de Assis

In this paper, we construct new families of classical convolutional codes (CCC's) and new families of quantum convolutional codes (QCC's). The CCC's are derived from (block) algebraic geometry (AG) codes. Furthermore, new families of CCC's are constructed by applying the techniques of puncturing, extending, expanding, and by the direct product code construction applied to AG codes. In addition, utilizing the new CCC's constructed here, we obtain new families of QCC's. The parameters of these new codes are good. More precisely, in the classical case, a family of almost near maximum distance separable (MDS) codes is presented; in the quantum case, we construct a family of MDS (optimal) quantum convolutional codes.

arXiv:1612.07157IEEE Transactions on CommunicationsOct 12, 20189 citations

Quantum Codes From Concatenated Algebraic-Geometric Codes

H. Chen, San Ling, Chaoping Xing

We apply Steane's enlargement of the Calderbank-Shor-Steane (CSS) codes and additive codes over F/sub 4/ to concatenated algebraic-geometric codes to construct many good quantum codes with fewer restrictions on the parameters compared to some known quantum codes. Some of the quantum codes we have constructed are either optimal or have parameters as good as the best known codes, while some have parameters better than those obtained from other known constructions.

IEEE Transactions on Information TheoryJul 19, 200571 citations

Entanglement-Assisted Quantum Codes From Algebraic Geometry Codes

Francisco Revson F. Pereira, Ruud Pellikaan, Giuliano G. La Guardia +1 more

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.

arXiv:1907.06357IEEE Transactions on Information TheorySep 16, 20213 citations

Algebraic decoding for classes of quantum codes

Yannick Saouter, Gilles Burel

This paper describes a technique to perform the decoding of quantum Calderbank-Steane-Shor codes built from self-orthogonal classical binary codes possessing algebraic decoding procedures. This technique gives a viable alternative to syndrome decoding for codes with large minimum distance. The cases of Reed-Muller and BCH codes are examined. The new method is compared with syndrome decoding on illustrative examples. The asymptotic cost of both methods is also given. Index Terms-quantum code, algebraic decoding, CSS quantum codes.

IEEE International Mediterranean Conference on Communications and Networking (MeditCom)Jul 8, 20240 citations

Classical self-orthogonal codes and their applications to quantum codes

Lingfei Jin

After the pioneering work of Shor and Steane, we are able to establish links between quantum codes and classical codes with certain self-orthogonality. Therefore, constructing classical self-orthogonal codes with small dimension and large dual distance becomes our centre point due to the interesting application to quantum codes. \n \n \n \nOne of the most interesting and useful families of classical codes are MDS codes. We will show a systematic construction for classical Hermitian self-orthogonal MDS codes through generalized Reed-Solomn codes. Afterwards, new families of quantum MDS codes can be produced. \n \n \n \nClassical BCH codes would be another good choice for self-orthogonal codes. We show the dual containment through polynomial evaluations, simply by choosing suitable cyclotomic cosets. As a result, quantum codes with good parameters can be derived. \n \n \n \nAnother good candidate of classical codes to investigate for dual containment comes from the class of \n \nalgebraic geometry codes. Instead of testing for dual containment directly, this thesis explores an equivalent condition for the existence of self-orthogonal AG codes. \n \n \n \nAfter various constructions for quantum codes, this thesis ends with a study on bounds for estimating the parameters of codes.

Jan 1, 20130 citations

Quantum Algebraic Geometry Codes

Zhengyi Han

Quantum error correction is an essential aspect of quantum information theory, providing protection for quantum states against noise and decoherence. This thesis investigates the construction of quantum error correction codes derived from classical algebraic geometry (AG) codes. We present two distinct construction techniques, highlighting the flexibility and self-orthogonality of AG codes, and demonstrate their ability to produce asymptotically good quantum codes. Additionally, we explore strategies to fine-tune the parameters of classical AG codes, ensuring they possess the desired properties for quantum code construction. This work serves as a comprehensive guide to the fundamental concepts and common methodologies underlying quantum algebraic geometry codes.

Dec 1, 20240 citations