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].
Protection
These models realize local topological order (LTO) [6].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 [7].Cousins
- 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 [8,9].
- XS stabilizer code— Abelian TQD models for the groups \(\mathbb{Z}_2^k\) can be realized as XS stabilizer codes [10]. Upon gauging some symmetries [8,11,12,12], a Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [13–15].
Member of code lists
Primary Hierarchy
Parents
TQD models are defined on group-valued qudits.
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 \(D^{\omega}(G)\) [16; pg. 41]. TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory [2; Sec. IX].
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].
Twisted quantum double (TQD) code
Children
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.
The anyon theory corresponding to a quantum-double code is a TQD with trivial cocycle.
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 [17].
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, Z. Komargodski, K. Ohmori, S. Seifnashri, and S.-H. Shao, “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]
- S. X. Cui, C. Galindo, and D. Romero, “Twisted Kitaev quantum double model as local topological order”, (2025) arXiv:2411.08675
- [7]
- 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
- [8]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [9]
- X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
- [10]
- X. Ni, O. Buerschaper, and M. Van den Nest, “A non-commuting stabilizer formalism”, Journal of Mathematical Physics 56, (2015) arXiv:1404.5327 DOI
- [11]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [12]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [13]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [14]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [15]
- L. Lootens, B. Vancraeynest-De Cuiper, N. Schuch, and F. Verstraete, “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
- [16]
- 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
- [17]
- T. D. Ellison, Y.-A. Chen, A. Dua, W. Shirley, N. Tantivasadakarn, and D. J. Williamson, “Pauli Stabilizer Models of Twisted Quantum Doubles”, PRX Quantum 3, (2022) arXiv:2112.11394 DOI
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.