GKP码的奇妙世界
The fabulous world of GKP codes
AI总结：
本文以GKP码为核心，系统梳理其无限维稳定子结构、平移对称性及与数学主题的联系，旨在阐明其编码理论与实用性质并为后续研究奠基。
摘要：
量子纠错是量子技术发展中的一个必要组成部分。其研究主题是探索将量子希尔伯特空间嵌入物理系统的方法，使该子空间能够抵御物理系统中的微小缺陷。这项任务极其复杂：一方面，这是因为可供使用的可能物理系统种类繁多，且具有不同的结构；另一方面，每一种物理环境也都伴随着自身需要加以防护的缺陷。在技术雄心所带来的复杂性推动下，量子纠错研究已经发展成一个庞大的研究领域，涵盖从利用单个光子工程化小型系统，到创造宏观拓扑物相以及复杂涌现物理模型等内容。量子纠错中的一种典型工具是稳定子形式体系，它通过施加对称性来驯服量子系统。Gottesman-Kitaev-Preskill（GKP）码是一种稳定子码，通过赋予无限维希尔伯特空间平移对称性，在其中创建一个逻辑子空间。尽管在实践中，希尔伯特空间的无限性以及平移对称群的无限性被认为是实现上的障碍，但在理论上，恰恰是这些特征使GKP码理论格外丰富、性质良好，并且与数学中引人入胜的主题紧密相连。本论文的目的正是探索这些联系：利用GKP码丰富的数学基础，理解其编码理论性质和实际性质，并为未来研究提供基础。在这一旅程中，我们透过GKP码这面镜子，发现量子纠错如何融入一个奇妙的数学世界，并围绕可能的研究方向阐述一系列梦想。
英文摘要：
Quantum error correction is an essential ingredient in the development of quantum technologies. Its subject is to investigate ways to embed quantum Hilbert spaces into a physical system such that this subspace is robust against small imperfections in the physical systems. This task is exceedingly complex: for one, this is due to the vast diversity of possible physical systems with different structure to use. For another, every physical setting also comes with its own imperfections that need to be protected against. Bred by the complexity of a technological ambition, research on quantum error correction has developed into a large field of research that ranges from engineering of small systems with a single photon to the creation of macroscopic topological phases of matter and models of complex emergent physics. A quintessential tool in quantum error correction is the stabilizer formalism, which tames quantum systems by enforcing symmetries. A Gottesman-Kitaev-Preskill (GKP) code is a stabilizer code that creates a logical subspace within an infinite dimensional Hilbert space by endowing it with translational symmetries. While in practice the infinitude of the Hilbert space, as well as the infinitude of the translational symmetry group are considered as obstacles for implementation, in theory these are precisely the features that make the theory of GKP codes particularly rich, well behaved and well-connected to fascinating topics in mathematics. The purpose of this thesis is to explore these connections: to understand the coding theoretic and practical properties of GKP codes, utilizing its rich mathematical foundation, and to provide a foundation for future research. Along this journey we discover -- through the looking glass of GKP codes -- how quantum error correction fits into a fabulous mathematical world and formulate a series of dreams about possible directions of research.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
