Multi-fusion string-net code[1]
Description
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).Cousin
- 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].
Member of code lists
Primary Hierarchy
Parents
Enriched string-net codes realize 2D topological phases based on unitary multi-fusion categories.
Multi-fusion string-net codes form eigenspaces of frustration-free commuting projector Hamiltonians.
Multi-fusion string-net codes form eigenspaces of frustration-free commuting projector Hamiltonians.
Multi-fusion string-net code
Children
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.
References
- [1]
- L. Chang, M. Cheng, S. X. Cui, Y. Hu, W. Jin, R. Movassagh, P. Naaijkens, Z. Wang, and A. Young, “On enriching the Levin–Wen model with symmetry”, Journal of Physics A: Mathematical and Theoretical 48, 12FT01 (2015) arXiv:1412.6589 DOI
- [2]
- O. Buerschaper, M. Christandl, L. Kong, and M. Aguado, “Electric–magnetic duality of lattice systems with topological order”, Nuclear Physics B 876, 619 (2013) arXiv:1006.5823 DOI
- [3]
- L. Chang, “Kitaev models based on unitary quantum groupoids”, Journal of Mathematical Physics 55, (2014) arXiv:1309.4181 DOI
- [4]
- P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories”, (2017) arXiv:math/0203060
- [5]
- A. Molnar, A. R. de Alarcón, J. Garre-Rubio, N. Schuch, J. I. Cirac, and D. Pérez-García, “Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states”, (2022) arXiv:2204.05940
- [6]
- Z. Jia, S. Tan, D. Kaszlikowski, and L. Chang, “On Weak Hopf Symmetry and Weak Hopf Quantum Double Model”, Communications in Mathematical Physics 402, 3045 (2023) arXiv:2302.08131 DOI
Page edit log
- Victor V. Albert (2022-12-05) — most recent
Cite as:
“Multi-fusion string-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/enriched_string_net