Multi-fusion string-net code[1] 


Family of codes resulting from the string-net construction but whose input is a unitary multi-fusion category (as opposed to a unitary fusion category).



  • Groupoid toric code — Groupoid toric-code categories are unitary multi-fusion categories based on matrix algebras [1], so groupoid toric codes can equivalently be formulated as multi-fusion string-net codes.
  • String-net code


  • Hopf-algebra quantum-double code — Extending the Hopf algebra quantum-double construction to a weak Hopf algebra construction yields an alternative formulation [3][2; Fig. 1] for realizing multi-fusion string-net topological orders because of the relationship between representations of weak Hopf algebras and multi-fusion categories [4]. Tensor network constructions can be done for either formulation [5,6].


L. Chang et al., “On enriching the Levin–Wen model with symmetry”, Journal of Physics A: Mathematical and Theoretical 48, 12FT01 (2015) arXiv:1412.6589 DOI
O. Buerschaper et al., “Electric–magnetic duality of lattice systems with topological order”, Nuclear Physics B 876, 619 (2013) arXiv:1006.5823 DOI
L. Chang, “Kitaev models based on unitary quantum groupoids”, Journal of Mathematical Physics 55, (2014) arXiv:1309.4181 DOI
P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories”, (2017) arXiv:math/0203060
A. Molnar et al., “Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states”, (2022) arXiv:2204.05940
Z. Jia et al., “On Weak Hopf Symmetry and Weak Hopf Quantum Double Model”, Communications in Mathematical Physics 402, 3045 (2023) arXiv:2302.08131 DOI
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Zoo Code ID: enriched_string_net

Cite as:
“Multi-fusion string-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.
@incollection{eczoo_enriched_string_net, title={Multi-fusion string-net code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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Cite as:

“Multi-fusion string-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022.