Pauli topological subsystem codes from Abelian anyon theories

• URL: http://arxiv.org/abs/2211.03798v1
• Date: Mon, 7 Nov 2022 19:00:01 GMT
• Title: Pauli topological subsystem codes from Abelian anyon theories
• Authors: Tyler D. Ellison, Yu-An Chen, Arpit Dua, Wilbur Shirley, Nathanan Tantivasadakarn, and Dominic J. Williamson
• Abstract summary: We construct Pauli topological subsystem codes characterized by arbitrary two-dimensional Abelian anyon theories. Our work both extends the classification of two-dimensional Pauli topological subsystem codes to systems of composite-dimensional qudits.
• Score: 2.410842777583321
• License: http://creativecommons.org/licenses/by/4.0/
• 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.

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