## Description

Code whose codewords realize 2D gapped topological order defined on qudits valued in a Hopf algebra \(H\). The code Hamiltonian is an generalization [1,2] of the quantum double model from group algebras to Hopf algebras, as anticipated by Kitaev [3]. Boundaries of these models have been examined [4,5].

## Parent

- String-net code — String-net model ground states reduce to Hopf-algebra quantum-double ground states for categories \(\text{Rep}(H)\), where \(H\) is a Hopf algebra [2].

## Child

- Quantum-double code — Hopf-algebra quantum-double codes reduce to quantum-double codes when the Hopf algebra is a group algebra. Quantum-double codes for non-Abelian groups \(G\) are dual to Hopf-algebra quantum-double codes for Hopf algebras based on \(\text{Rep}(G)\) under the Tannaka-Krein duality [6][7; Fig. 1].

## Cousins

- Multi-fusion string-net code — Extending the Hopf algebra quantum-double construction to a weak Hopf algebra construction yields an alternative formulation [8][7; Fig. 1] for realizing multi-fusion string-net topological orders because of the relationship between representations of weak Hopf algebras and multi-fusion categories [9]. Tensor network constructions can be done for either formulation [10,11].
- Modular-qudit surface code — The modular-qudit surface code can be generalized to a Hopf-algebra quantum-double code whose ground states remain the same but whose excitations are based on quasitriangular semisimple Hopf algebras of \(\mathbb{Z}_q\) [12].
- Hopf-algebra cluster-state code — Both Hopf-algebra quantum-double and Hopf-algebra cluster-state codes are defined on qudits valued in a Hopf algebra.

## References

- [1]
- O. Buerschaper et al., “A hierarchy of topological tensor network states”, Journal of Mathematical Physics 54, (2013) arXiv:1007.5283 DOI
- [2]
- B. Balsam and A. Kirillov Jr, “Kitaev’s Lattice Model and Turaev-Viro TQFTs”, (2012) arXiv:1206.2308
- [3]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [4]
- Z. Jia, D. Kaszlikowski, and S. Tan, “Boundary and domain wall theories of 2d generalized quantum double model”, Journal of High Energy Physics 2023, (2023) arXiv:2207.03970 DOI
- [5]
- A. Cowtan and S. Majid, “Algebraic Aspects of Boundaries in the Kitaev Quantum Double Model”, (2022) arXiv:2208.06317
- [6]
- O. Buerschaper and M. Aguado, “Mapping Kitaev’s quantum double lattice models to Levin and Wen’s string-net models”, Physical Review B 80, (2009) arXiv:0907.2670 DOI
- [7]
- O. Buerschaper et al., “Electric–magnetic duality of lattice systems with topological order”, Nuclear Physics B 876, 619 (2013) arXiv:1006.5823 DOI
- [8]
- L. Chang, “Kitaev models based on unitary quantum groupoids”, Journal of Mathematical Physics 55, (2014) arXiv:1309.4181 DOI
- [9]
- P. Etingof, D. Nikshych, and V. Ostrik, “On fusion categories”, (2017) arXiv:math/0203060
- [10]
- A. Molnar et al., “Matrix product operator algebras I: representations of weak Hopf algebras and projected entangled pair states”, (2022) arXiv:2204.05940
- [11]
- 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
- [12]
- A. Conlon, D. Pellegrino, and J. K. Slingerland, “Modified toric code models with flux attachment from Hopf algebra gauge theory”, Journal of Physics A: Mathematical and Theoretical 56, 295302 (2023) arXiv:2210.07909 DOI

## Page edit log

- Victor V. Albert (2024-04-27) — most recent
- Laurens Lootens (2024-04-27)

## Cite as:

“Hopf-algebra quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/hopf_quantum_double