Pauli topological subsystem codes from Abelian anyon theories

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories -- this includes non-modular anyon theories and those without a Lagrangian subgroup. We exemplify the construction with topological subsystem codes based on the non-modular Z₄⁽¹⁾ anyon theory and the chiral semion theory, both of which cannot be captured by topological stabilizer codes. The construction proceeds by "gauging out" certain anyon types of a topological stabilizer code. This amounts to defining a gauge group generated by the stabilizer group of the topological stabilizer code and a set of anyonic string operators for the anyon types that are gauged out. The resulting topological subsystem code is characterized by an anyon theory that is a proper subset of the anyons of the topological stabilizer code. We thereby show that every Abelian anyon theory is a subtheory of a stack of toric codes and a certain family of twisted quantum doubles that generalize the double semion anyon theory. Our work thus extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and shows that the classification is at least as rich as that of two-dimensional Abelian anyon theories. We further prove a number of general statements about the logical operators of translation invariant topological subsystem codes and define their associated anyon theories in terms of higher-form symmetries.