Pauli topological subsystem codes from Abelian anyon theories

TL;DR

This paper introduces a method to construct Pauli topological subsystem codes from arbitrary Abelian anyon theories, extending the classification to systems with complex braiding and boundary properties, including non-stabilizer codes.

Contribution

It generalizes the construction of topological subsystem codes to include composite qudits and complex anyon theories, demonstrating that all Abelian anyon theories can be embedded within such codes.

Findings

Constructed codes for 4D qudits based on complex anyon theories.

Showed all Abelian anyon theories are subtheories of known topological models.

Provided insights into logical operators and higher-form symmetries in these codes.

Abstract

We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories--this includes anyon theories with degenerate braiding relations and those without a gapped boundary to the vacuum. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits and establishes that the classification is at least as rich as that of Abelian anyon theories. We exemplify the construction with topological subsystem codes defined on four-dimensional qudits based on the $\mathbb{Z}_4^{(1)}$ anyon theory with degenerate braiding relations 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 containing 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. 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.