Alternative names: 2D Dijkgraaf-Witten gauge theory code.
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 3-cocycle \(\omega\in H^3( G, U(1) )\) [2,4,5]. Canonical TQD models [2] are defined on group-valued qudits.
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 Drinfeld double) \(D^{\omega}(G)\). Gapped boundaries of the models are classified by a subgroup \(K \subseteq G\) and a particular two-cochain [6].
Protection
These models realize local topological order (LTO) [7].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 [8].Cousins
- Symmetry-protected topological (SPT) code— A TQD code Hamiltonian can be obtained from a 2D particular SPT model by gauging [9–18] the model’s background gauge. The same group and cocycle data classifies both 2D SPTs and TQDs [9,19].
- Twisted quantum triple (TQT) code— The TQT model can be thought of as a 3D version of the TQD model.
Member of code lists
Primary Hierarchy
Parents
Restricting Dijkgraaf-Witten gauge theory to a 2D manifold reproduces the phase of the TQD model [4]. The Drinfeld 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 \(D^{\omega}(G)\) [20; pg. 41]. TQD codewords are gauge-invariant boundary states of a 3D Dijkgraaf-Witten theory [2; Sec. IX].
String-net models realize TQDs for categories \(\text{Vec}^{\omega}G\), where \(G\) is a finite group and \(\omega\) is a 3-cocycle on \(G\). 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 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-I, Type-II, or Type-III 3-cocycle. Abelian TQDs with Type-I and -II cocycles account for all 2D Abelian topological orders that admit gapped boundaries [21].
The ground-state subspace of the hexagonal \(CZ\) code realizes the topological order of the Type-III \(G=\mathbb{Z}^3_2\) Abelian TQD model [22,23], which is the same topological order as the \(G=D_4\) quantum double [24]. There is a constant-depth circuit implementing a transversal logical \(T\) gate via an emergent automorphism symmetry of the underlying \(\mathbb{D}_4\) topological order [25].
References
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- 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]
- 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
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- 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
- [7]
- S. X. Cui, C. Galindo, and D. Romero, “Twisted Kitaev quantum double model as local topological order”, (2025) arXiv:2411.08675
- [8]
- 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
- [9]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [10]
- J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
- [11]
- S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
- [12]
- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
- [13]
- 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
- [14]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [15]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [16]
- K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
- [17]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
- [18]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
- [19]
- X. Chen, Z.-C. Gu, Z.-X. Liu, and X.-G. Wen, “Symmetry protected topological orders in interacting bosonic systems”, (2013) arXiv:1301.0861
- [20]
- 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
- [21]
- A. Kapustin and N. Saulina, “Topological boundary conditions in abelian Chern–Simons theory”, Nuclear Physics B 845, 393 (2011) arXiv:1008.0654 DOI
- [22]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [23]
- P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory and Transversal Non-Clifford Gate Beyond the n \({}^{\text{1/3}}\) Distance Barrier”, PRX Quantum 6, (2025) arXiv:2405.11719 DOI
- [24]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [25]
- R. Kobayashi, G. Zhu, and P.-S. Hsin, “Clifford Hierarchy Stabilizer Codes: Transversal Non-Clifford Gates and Magic”, (2025) arXiv:2511.02900
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