Pauli topological subsystem codes from Abelian anyon theories

Tyler D. Ellison1, Yu-An Chen2, Arpit Dua3, Wilbur Shirley4, Nathanan Tantivasadakarn5,6, and Dominic J. Williamson7

1Department of Physics, Yale University, New Haven, CT 06511, USA
2Department of Physics, Condensed Matter Theory Center, Joint Quantum Institute, and Joint Center for Quantum Information and Computer Science, University of Maryland, College Park, MD 20742, USA
3Department of Physics and Institute for Quantum Information and Matter, California Institute of Technology, Pasadena, CA 91125, USA
4School of Natural Sciences, Institute for Advanced Study, Princeton, NJ 08540, USA
5Walter Burke Institute for Theoretical Physics and Department of Physics, California Institute of Technology, Pasadena, CA 91125, USA
6Department of Physics, Harvard University, Cambridge, MA 02138, USA
7Centre for Engineered Quantum Systems, School of Physics, University of Sydney, Sydney, New South Wales 2006, Australia

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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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