## 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. Codewords are gauge-invariant boundary states of a Dijkgraaf-Witten theory [2; Sec. IX].

A code Hamiltonian can be obtained from a 2D model with symmetry-protected topological (SPT) order by gauging the model's symmetry. The same group and cocycle data classifies both 2D SPTs and TQDs [4].

## Encoding

For any solvable group \(G\), ground-state preparation can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [5].

## Parents

- Group-based quantum code
- String-net code — String-net models reduce to TQDs for categories \(\text{Vec}_G^\omega\), 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) [6].
- Abelian TQD stabilizer code — The anyon theory corresponding to (Abelian) TQD codes is defined by an (Abelian) group and a Type III cocycle.

## 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]
- X. Chen et al., “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
- [5]
- 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
- [6]
- M. Tomba et al., “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440

## 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.