## Description

Code whose codewords realize a 2D topological order rendered by a Chern-Simons topological field theory. The corresponding anyon theory is defined by a finite group \(G\) and a Type-III group cocycle \(\omega\), but can also be described in a category theoretic way [4].

Logical dimension is determined by the genus of the underlying surface (for closed surfaces), types of boundaries (for open surfaces), and any twist defects present. Excitations are described by the twisted quantum double (a.k.a. twisted Drinfield double) \(D^{\omega}(G)\). Gapped boundaries of the models are classified by a subgroup \(K \subseteq G\) and a particular two-cochain [5].

## Encoding

## Parents

- Dijkgraaf-Witten gauge theory code — Restricting Dijkgraaf-Witten gauge theory to a 2D manifold reproduces the phase of the TQD model [4]. The Drinfield center of the category \(\text{Vec}^{\omega}(G)\) is used to describe bulk excitations of 3D Dijkgraaf-Witten models, and this center is equivalent to the twisted quantum double \(D^{\omega}(G)\) [7; pg. 41]. TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory [2; Sec. IX].
- String-net code — String-net models reduce to TQDs for categories \(\text{Vec}^{\omega}G\), where \(G\) is a finite group, and \(\omega\) is a Type III cocycle. There is a duality between a large class of string–net models and certain TQD models [2].

## Children

- Generalized 2D color code — The anyon theory corresponding to a generalized color code is a trivial-cocycle TQD associated with the group \(G \times G/[G,G]\), where \(G\) is any finite group.
- Quantum-double code — The anyon theory corresponding to a quantum-double code is a TQD with trivial cocycle. These models realize local topological order (LTO) [8].
- Abelian TQD stabilizer code — The anyon theory corresponding to (Abelian) TQD codes is defined by an (Abelian) group and a Type III cocycle. Abelian TQDs realize all modular gapped Abelian topological orders [9].

## Cousin

- Symmetry-protected topological (SPT) code — A TQD code Hamiltonian can be obtained from a 2D particular SPT model by promoting the model's background gauge field to a dynamical gauge field. The same group and cocycle data classifies both 2D SPTs and TQDs [10,11].

## References

- [1]
- R. Dijkgraaf, V. Pasquier, and P. Roche, “Quasi hope algebras, group cohomology and orbifold models”, Nuclear Physics B - Proceedings Supplements 18, 60 (1991) DOI
- [2]
- Y. Hu, Y. Wan, and Y.-S. Wu, “Twisted quantum double model of topological phases in two dimensions”, Physical Review B 87, (2013) arXiv:1211.3695 DOI
- [3]
- J. Kaidi et al., “Higher central charges and topological boundaries in 2+1-dimensional TQFTs”, SciPost Physics 13, (2022) arXiv:2107.13091 DOI
- [4]
- D. Naidu and D. Nikshych, “Lagrangian Subcategories and Braided Tensor Equivalences of Twisted Quantum Doubles of Finite Groups”, Communications in Mathematical Physics 279, 845 (2008) arXiv:0705.0665 DOI
- [5]
- A. Bullivant, Y. Hu, and Y. Wan, “Twisted quantum double model of topological order with boundaries”, Physical Review B 96, (2017) arXiv:1706.03611 DOI
- [6]
- N. Tantivasadakarn, A. Vishwanath, and R. Verresen, “Hierarchy of Topological Order From Finite-Depth Unitaries, Measurement, and Feedforward”, PRX Quantum 4, (2023) arXiv:2209.06202 DOI
- [7]
- A. Bullivant and C. Delcamp, “Tube algebras, excitations statistics and compactification in gauge models of topological phases”, Journal of High Energy Physics 2019, (2019) arXiv:1905.08673 DOI
- [8]
- M. Tomba et al., “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440
- [9]
- T. D. Ellison et al., “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
- [10]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [11]
- X. Chen et al., “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861

## Page edit log

- Victor V. Albert (2023-04-06) — most recent

## Cite as:

“Twisted quantum double (TQD) code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/tqd

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/groups/topological/tqd.yml.