量子码的构造

项目名称： 量子码的构造

项目编号： No.11501117

项目类型： 青年科学基金项目

立项/批准年度： 2016

项目学科： 数理科学和化学

项目作者： 金玲飞

作者单位： 复旦大学

项目金额： 18万元

中文摘要： 类似于经典的通信和计算机，量子纠错码的研究对于量子通信和量子计算机是个至关重要的课题。量子MDS码是一类重要的最优码，具有很强的实际应用背景。特别是构造维数为1的量子MDS码是研究量子密钥共享的关键问题之一。更一般地，对于给定r，构造码长尽可能长的参数为[[n,n-2d+2,d-r]]的量子码具有很大的挑战性，尤其是对于r较小的情况。本项目将利用数论，代数几何等知识研究几类重要的经典码的自正交性质，从而构造出新的量子MDS码及一般的量子码。具体的说，我们通过给出Reed-Solomon 码的Hermitan 自正交条件，从而构造出一些新的量子MDS码。众所周知，目前仍然没有较好的方法能将Reed-Solomon 码的自正交性质推广到代数曲线。本项目通过研究椭圆曲线群结构及亏格较小的代数曲线上的微分来给出代数几何码自正交性质并构造出所需的量子码。最后通过代数几何码的性质改进现有的渐进界结果。

中文关键词： 代数几何码；自正交性；MDS码；量子码

英文摘要： As in classical communication and computation, the study of quantum error correcting codes is essential for quantum communication and quantum computing. Quantum MDS codes are an important class of optimal codes for their wide applications and nice mathematical structures. For instance, the construction of quantum MDS codes with dimension 1 plays a key role on quantum secret sharing scheme. Generally, for given r, we are interested in constructing [[n,n-2d+2,d-r]] quantum codes with length n as large as possible, especially for small r. In this project, we will mainly use tools from mathematical subjects like number theory, algebraic geometry, etc., to investigate the self-orthogonality of certain families of classical codes, and finally construct some new quantum MDS codes and even more general small defect quantum codes. More specifically, by exploring the Hermitian self-orthogonality of generalized Reed-Solomon codes, we will construct some new quantum MDS codes. However, in the literature, there is no existing good method to generalize the self-orthogonality of Reed-Solomon codes to algebraic curves. In this project, by studying the group structure of elliptic curves and the differential of algebraic curves with small genus, we plan to obtain the self-orthogonality of algebraic geometry codes and then construct good quantum codes. Finally, we will use algebraic geometry codes and their good self-orthogonality to improve the known asymptotic bounds, which is another central problem in quantum coding theory.

英文关键词： algebraic geometry code;self-orthogonality;MDS code;quantum code