标题： 优良LDPC码的物理性质II. 乘积构造 显示英文标题

标题： The Physics of (good) LDPC Codes II. Product constructions

摘要： 我们从物理学的角度继续研究经典和量子低密度奇偶校验（LDPC）码。 我们专注于构造性方法，并制定了一种通用框架，用于系统地在欧几里得和非欧几里得图上构建具有各种特性的码。 这些码可以作为物质相的固定点极限。为了构建我们的工具，我们根据物理原理（如对称性和冗余性）拆解了编码文献中的各种乘积构造，引入了一种新的立方乘积，并结合了第一部分中介绍的规范化和希格斯化思想。 通过展示从一维伊辛模型出发，我们可以系统地生成非常广泛的经典和量子物质相，包括I型和II型分数量子自旋液体以及具有广义对称性的SPT相，从而证明了这种方法在有限欧几里得维度中的实用性。 我们还利用平衡积构建了新的欧几里得模型，包括一个拓扑序由平移对称性增强的模型，以及另一个由分形自旋液体的激发与 toric 码的激发组合而成的奇异分数量子自旋液体模型，从而产生独特的移动约束。 超越欧几里得模型，我们回顾了良好qLDPC码和经典局部可测试码的现有构造，并从乘积构造的角度详细阐述了第一部分中讨论的量子码距离与经典能量势垒之间的关系。 显示英文摘要

摘要： We continue the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective. We focus on constructive approaches and formulate a general framework for systematically constructing codes with various features on generic Euclidean and non-Euclidean graphs. These codes can serve as fixed-point limits for phases of matter. To build our machinery, we unpack various product constructions from the coding literature in terms of physical principles such as symmetries and redundancies, introduce a new cubic product, and combine these products with the ideas of gauging and Higgsing introduced in Part I. We illustrate the usefulness of this approach in finite Euclidean dimensions by showing that using the one-dimensional Ising model as a starting point, we can systematically produce a very large zoo of classical and quantum phases of matter, including type I and type II fractons and SPT phases with generalized symmetries. We also use the balanced product to construct new Euclidean models, including one with topological order enriched by translation symmetry, and another exotic fracton model whose excitations are formed by combining those of a fractal spin liquid with those of a toric code, resulting in exotic mobility constraints. Moving beyond Euclidean models, we give a review of existing constructions of good qLDPC codes and classical locally testable codes and elaborate on the relationship between quantum code distance and classical energy barriers, discussed in Part I, from the perspective of product constructions.