Pauli Topological Subsystem Codes From Abelian Anyon Theories

Tyler D. Ellison¹, Yu-An Chen², Arpit Dua³, Wilbur Shirley⁴, Nathanan Tantivasadakarn^5,6, 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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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.

► BibTeX data

@article{Ellison2023paulitopological, doi = {10.22331/q-2023-10-12-1137}, url = {https://doi.org/10.22331/q-2023-10-12-1137}, title = {Pauli topological subsystem codes from {A}belian anyon theories}, author = {Ellison, Tyler D. and Chen, Yu-An and Dua, Arpit and Shirley, Wilbur and Tantivasadakarn, Nathanan and Williamson, Dominic J.}, journal = {{Quantum}}, issn = {2521-327X}, publisher = {{Verein zur F{“{o}}rderung des Open Access Publizierens in den Quantenwissenschaften}}, volume = {7}, pages = {1137}, month = oct, year = {2023}<br /> }

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Cited by

[1] Hector Bombin, Chris Dawson, Terry Farrelly, Yehua Liu, Naomi Nickerson, Mihir Pant, Fernando Pastawski, and Sam Roberts, “Fault-tolerant complexes”, arXiv:2308.07844, (2023).

[2] Tyler D. Ellison, Joseph Sullivan, and Arpit Dua, “Floquet codes with a twist”, arXiv:2306.08027, (2023).

[3] Li-Mei Chen, Tyler D. Ellison, Meng Cheng, Peng Ye, and Ji-Yao Chen, “Chiral Fibonacci spin liquid in a $mathbb{Z}_3$ Kitaev model”, arXiv:2302.05060, (2023).

[4] Jacob C. Bridgeman, Aleksander Kubica, and Michael Vasmer, “Lifting topological codes: Three-dimensional subsystem codes from two-dimensional anyon models”, arXiv:2305.06365, (2023).

[5] Arpit Dua, Nathanan Tantivasadakarn, Joseph Sullivan, and Tyler D. Ellison, “Engineering Floquet codes by rewinding”, arXiv:2307.13668, (2023).

[6] Dominic J. Williamson and Nouédyn Baspin, “Layer Codes”, arXiv:2309.16503, (2023).

[7] Po-Shen Hsin and Zhenghan Wang, “On topology of the moduli space of gapped Hamiltonians for topological phases”, Journal of Mathematical Physics 64 4, 041901 (2023).

[8] Daniel Bulmash, Oliver Hart, and Rahul Nandkishore, “Multipole groups and fracton phenomena on arbitrary crystalline lattices”, arXiv:2301.10782, (2023).

[9] Andreas Bauer, “Topological error correcting processes from fixed-point path integrals”, arXiv:2303.16405, (2023).

[10] Rahul Sarkar and Theodore J. Yoder, “The qudit Pauli group: non-commuting pairs, non-commuting sets, and structure theorems”, arXiv:2302.07966, (2023).

The above citations are from SAO/NASA ADS (last updated successfully 2023-10-13 03:20:12). The list may be incomplete as not all publishers provide suitable and complete citation data.

On Crossref’s cited-by service no data on citing works was found (last attempt 2023-10-13 03:20:11).

This Paper is published in Quantum under the Creative Commons Attribution 4.0 International (CC BY 4.0) license. Copyright remains with the original copyright holders such as the authors or their institutions.

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Pauli topological subsystem codes from Abelian anyon theories

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Pauli topological subsystem codes from Abelian anyon theories

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