This paper continues the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective, focusing on constructive approaches and formulating 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. The authors 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. They illustrate the usefulness of this approach in finite Euclidean dimensions by showing that using the one-dimensional Ising model as a starting point, they can systematically produce a very large zoo of classical and quantum phases of matter, including type I and type II fractions and SPT phases with generalized symmetries. They also use the balanced product to construct new Euclidean models, including one with topological order enriched by translation symmetry and another exotic fraction 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, they 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.
The paper discusses the physical role of graph vs. chain complex dimensionality, emphasizing that classical stabilizer codes are naturally associated with one-dimensional chain complexes, while quantum CSS codes are associated with two-dimensional chain complexes. The authors introduce a new product construction, the cubic product, which takes three classical codes as input and produces a code with both local redundancies and subsystem symmetries. Combining these product constructions with other ingredients, including the duality maps from Part I, yields an entire machinery that can be used to systematically build classical and quantum models with increasingly intricate properties. The paper also discusses the relationship between locally testable classical codes and good quantum codes, and the connection between quantum code distance and classical energy barriers. The authors conclude by emphasizing the importance of the features (symmetries, redundancies, etc.) of the underlying classical codes and the need to find codes with particular features. They also discuss the power of the "code factory" in generating a vast array of known gapped phases of matter and elucidating a large web of connections between these different models. The paper also discusses the construction of two novel stabilizer models in two and three spatial dimensions, which are described as non-trivial symmetry-enriched topological (SET) phases and a fraction phase with exotic mobility properties. The authors also discuss the relationship between good qLDPC codes and classical LTCs in the context of product constructions.This paper continues the study of classical and quantum low-density parity check (LDPC) codes from a physical perspective, focusing on constructive approaches and formulating 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. The authors 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. They illustrate the usefulness of this approach in finite Euclidean dimensions by showing that using the one-dimensional Ising model as a starting point, they can systematically produce a very large zoo of classical and quantum phases of matter, including type I and type II fractions and SPT phases with generalized symmetries. They also use the balanced product to construct new Euclidean models, including one with topological order enriched by translation symmetry and another exotic fraction 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, they 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.
The paper discusses the physical role of graph vs. chain complex dimensionality, emphasizing that classical stabilizer codes are naturally associated with one-dimensional chain complexes, while quantum CSS codes are associated with two-dimensional chain complexes. The authors introduce a new product construction, the cubic product, which takes three classical codes as input and produces a code with both local redundancies and subsystem symmetries. Combining these product constructions with other ingredients, including the duality maps from Part I, yields an entire machinery that can be used to systematically build classical and quantum models with increasingly intricate properties. The paper also discusses the relationship between locally testable classical codes and good quantum codes, and the connection between quantum code distance and classical energy barriers. The authors conclude by emphasizing the importance of the features (symmetries, redundancies, etc.) of the underlying classical codes and the need to find codes with particular features. They also discuss the power of the "code factory" in generating a vast array of known gapped phases of matter and elucidating a large web of connections between these different models. The paper also discusses the construction of two novel stabilizer models in two and three spatial dimensions, which are described as non-trivial symmetry-enriched topological (SET) phases and a fraction phase with exotic mobility properties. The authors also discuss the relationship between good qLDPC codes and classical LTCs in the context of product constructions.