Quantum-double code[1]
Description
Group-GKP stabilizer code whose codewords realize 2D modular gapped topological order defined by a finite group \(G\). The code's generators are few-body operators associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit of dimension \( |G| \) located at each edge of the tesselation).
The physical Hilbert space has dimension \( |G|^E \), where \( E \) is the number of edges in the tessellation. The dimension of the code space is the number of orbits of the conjugation action of \( G \) on \( \text{Hom}(\pi_1(\Sigma),G) \), the set of group homomorphisms from the fundamental group of the surface \( \Sigma \) into the finite group \( G \) [2]. When \( G \) is Abelian, the formula for the dimension simplifies to \( |G|^{2g} \), where \( g \) is the genus of the surface \( \Sigma \).
The codespace is the ground-state subspace of the quantum double model Hamiltonian, while local excitations are characterized by anyons. Different types of anyons are labeled by irreducible representations of the group's quantum double algebra, \(D(G)\) (a.k.a. Drinfield center) [3,4]. Not all isomorphic non-Abelian groups give rise to different quantum doubles [5].
For non-Abelian groups, alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [6–8]. The fusion space of such non-Abelian anyons has dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.
Gapped boundaries of the models are classified by a subgroup \(K \subseteq G\) and a two-cocycle [3,9–11].
Protection
Error-correcting properties established in Ref. [2]. The code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [12]. These models realize local topological order (LTO) [13].Encoding
A depth-\(L^2\) circuit that grows the code out of a small patch on an \(L\times L\) square lattice using CMULT gates (i.e., "local moves") [14,15].For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [15,16] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [14,17].For any group \(G\) of nilpotency class two, states can be initialized with a single round of adaptive measurements [18].For any solvable group \(G\), ground-state preparation and anyon-pair creation can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [19,20] (see Ref. [21] for specific dihedral groups). Anyon-pair creation requires an adaptive circuit for any non-Abelian \(G\) [20].For non-solvable groups, states may not be preparable with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [18].Decoding
For any solvable group \(G\), topological charge measurements can be done with an adaptive constant-depth circuit with geometrically local gates and measurements throughout [20].Code Capacity Threshold
Behavior under particular \(X\)-type noise (namely, diffusion of an anyon that squares to the trivial anyon) is related to the phase diagram of a disordered \(D_4\) rotor model [22,23].Notes
See Ref. [24] for a review of gauge theory, which admits quantum-double topological phases.Cousins
- Hamiltonian-based code— Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the \([[4,1,1,2]]\) subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms [25].
- Subsystem QECC— Subsystem versions of quantum-double codes have been formulated [26].
- Two-gauge theory code— Restricting 2-gauge theory constructions to a 2D manifold and replacing the 2-group with a group reproduces the phase of the Kitaev quantum double model [27].
- \([[4,1,1,2]]\) Four-qubit subsystem code— Quantum double code Hamiltonians can be simulated, with the help of perturbation theory and the four-qubit subsystem code, by two-dimensional two-body Hamiltonians with non-commuting terms [25].
- Symmetry-protected topological (SPT) code— The \(Q\) quantum double model can be obtained by gauging [28–37] symmetries of a Type I and Type III \(\mathbb{Z}_2^3\) SPT [38,39].
- Abelian TQD code— A Type-III \(\mathbb{Z}_2^3\) TQD realizes the same topological order as the \(G=D_4\) quantum double model [40–42]. There is a sufficient condition for when a Type-III TQD can be realized as a quantum double model [43].
- Generalized 2D color code— Generalized color code for group \(G\) on the 4.8.8 lattice is equivalent to a \(G\) quantum double model and another \(G/[G,G]\) quantum double model defined using the Abelianization of \(G\).
- Kitaev surface code— A quantum-double model with \(G=\mathbb{Z}_2\) is the surface code. Non-stabilizer surface-code states can be prepared by augmenting the surface code with a quantum double model [44,45].
Primary Hierarchy
Parents
Quantum-double Hamiltonians can be expressed in terms of \(X\)- and \(Z\)-type operators of group-GKP codes; see [46; Sec. 3.3].
The anyon theory corresponding to a quantum-double code is a TQD with trivial cocycle.
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 [47][48; Fig. 1].
Quantum-double code
Children
The four group-qudit code is the smallest quantum double code.
The anyon theory corresponding to (Abelian) quantum double codes is defined by an (Abelian) group.
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [49,51–53][50; Sec. 5.3]. Galois-qudit color codes yield Abelian quantum-double codes with Abelian-group topological order via this decomposition.
A Galois qudit for \(q=p^m\) can be decomposed into a Kronecker product of \(m\) modular qudits [49,51–53][50; Sec. 5.3]. Galois-qudit surface codes yield Abelian quantum-double codes with \(GF(p^m)\cong \mathbb{Z}_p^m\) topological order via this decomposition.
References
- [1]
- A. Yu. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003) arXiv:quant-ph/9707021 DOI
- [2]
- S. X. Cui, D. Ding, X. Han, G. Penington, D. Ranard, B. C. Rayhaun, and Z. Shangnan, “Kitaev’s quantum double model as an error correcting code”, Quantum 4, 331 (2020) arXiv:1908.02829 DOI
- [3]
- S. Beigi, P. W. Shor, and D. Whalen, “The Quantum Double Model with Boundary: Condensations and Symmetries”, Communications in Mathematical Physics 306, 663 (2011) arXiv:1006.5479 DOI
- [4]
- A. Bols and S. Vadnerkar, “Classification of the Anyon Sectors of Kitaev’s Quantum Double Model”, Communications in Mathematical Physics 406, (2025) arXiv:2310.19661 DOI
- [5]
- D. Naidu, “Categorical Morita equivalence for group-theoretical categories”, (2006) arXiv:math/0605530
- [6]
- R. Walter Ogburn and J. Preskill, “Topological Quantum Computation”, Lecture Notes in Computer Science 341 (1999) DOI
- [7]
- C. Mochon, “Anyon computers with smaller groups”, Physical Review A 69, (2004) arXiv:quant-ph/0306063 DOI
- [8]
- J. K. Pachos, Introduction to Topological Quantum Computation (Cambridge University Press, 2012) DOI
- [9]
- H. Bombin and M. A. Martin-Delgado, “Family of non-Abelian Kitaev models on a lattice: Topological condensation and confinement”, Physical Review B 78, (2008) arXiv:0712.0190 DOI
- [10]
- 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
- [11]
- J. Huxford and S. H. Simon, “Excitations in the higher-lattice gauge theory model for topological phases. II. The (2+1) -dimensional case”, Physical Review B 108, (2023) arXiv:2204.05341 DOI
- [12]
- E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [13]
- M. Tomba, S. Wei, B. Hungar, D. Wallick, K. Kawagoe, C. Y. Chuah, and D. Penneys, “Boundary algebras of the Kitaev Quantum Double model”, (2023) arXiv:2309.13440
- [14]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
- [15]
- M. Aguado, “From entanglement renormalisation to the disentanglement of quantum double models”, Annals of Physics 326, 2444 (2011) arXiv:1101.0527 DOI
- [16]
- G. K. Brennen, M. Aguado, and J. I. Cirac, “Simulations of quantum double models”, New Journal of Physics 11, 053009 (2009) arXiv:0901.1345 DOI
- [17]
- R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
- [18]
- 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
- [19]
- N. Tantivasadakarn, R. Thorngren, A. Vishwanath, and R. Verresen, “Long-Range Entanglement from Measuring Symmetry-Protected Topological Phases”, Physical Review X 14, (2024) arXiv:2112.01519 DOI
- [20]
- S. Bravyi, I. Kim, A. Kliesch, and R. Koenig, “Adaptive constant-depth circuits for manipulating non-abelian anyons”, (2022) arXiv:2205.01933
- [21]
- R. Verresen, N. Tantivasadakarn, and A. Vishwanath, “Efficiently preparing Schrödinger’s cat, fractons and non-Abelian topological order in quantum devices”, (2022) arXiv:2112.03061
- [22]
- P. Sala and R. Verresen, “Stability and Loop Models from Decohering Non-Abelian Topological Order”, Physical Review Letters 134, (2025) arXiv:2409.12230 DOI
- [23]
- P. Sala, J. Alicea, and R. Verresen, “Decoherence and Wave-Function Deformation of D4 Non-Abelian Topological Order”, Physical Review X 15, (2025) arXiv:2409.12948 DOI
- [24]
- J. B. Kogut, “An introduction to lattice gauge theory and spin systems”, Reviews of Modern Physics 51, 659 (1979) DOI
- [25]
- C. G. Brell, S. T. Flammia, S. D. Bartlett, and A. C. Doherty, “Toric codes and quantum doubles from two-body Hamiltonians”, New Journal of Physics 13, 053039 (2011) arXiv:1011.1942 DOI
- [26]
- P. Kumar, “A Class of Quantum Double Subsystem Codes”, (2011) DOI
- [27]
- A. Bullivant, M. Calçada, Z. Kádár, P. Martin, and J. F. Martins, “Topological phases from higher gauge symmetry in3+1dimensions”, Physical Review B 95, (2017) arXiv:1606.06639 DOI
- [28]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [29]
- 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
- [30]
- 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
- [31]
- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
- [32]
- 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
- [33]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [34]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [35]
- 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
- [36]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
- [37]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
- [38]
- M. de W. Propitius and F. A. Bais, “Discrete gauge theories”, (1996) arXiv:hep-th/9511201
- [39]
- A. Lyons, C. F. B. Lo, N. Tantivasadakarn, A. Vishwanath, and R. Verresen, “Protocols for Creating Anyons and Defects via Gauging”, (2025) arXiv:2411.04181
- [40]
- M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
- [41]
- B. Yoshida, “Topological phases with generalized global symmetries”, Physical Review B 93, (2016) arXiv:1508.03468 DOI
- [42]
- 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
- [43]
- B. Li and G. Liu, “On Gauge Equivalence of Twisted Quantum Doubles”, (2024) arXiv:2408.09353
- [44]
- K. Laubscher, D. Loss, and J. R. Wootton, “Universal quantum computation in the surface code using non-Abelian islands”, Physical Review A 100, (2019) arXiv:1811.06738 DOI
- [45]
- R. Sajith, Z. Song, B. Roberts, V. Menon, and Y. Li, “Non-Clifford gates between stabilizer codes via non-Abelian topological order”, (2025) arXiv:2505.18265
- [46]
- V. V. Albert, D. Aasen, W. Xu, W. Ji, J. Alicea, and J. Preskill, “Spin chains, defects, and quantum wires for the quantum-double edge”, (2021) arXiv:2111.12096
- [47]
- 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
- [48]
- O. Buerschaper, M. Christandl, L. Kong, and M. Aguado, “Electric–magnetic duality of lattice systems with topological order”, Nuclear Physics B 876, 619 (2013) arXiv:1006.5823 DOI
- [49]
- A. Ashikhmin and E. Knill, “Nonbinary quantum stabilizer codes”, IEEE Transactions on Information Theory 47, 3065 (2001) DOI
- [50]
- A. Niehage, “Quantum Goppa Codes over Hyperelliptic Curves”, (2005) arXiv:quant-ph/0501074
- [51]
- D. Gottesman. Surviving as a quantum computer in a classical world (2024) URL
- [52]
- Q. T. Nguyen, “Good binary quantum codes with transversal CCZ gate”, (2024) arXiv:2408.10140
- [53]
- L. Golowich and V. Guruswami, “Asymptotically Good Quantum Codes with Transversal Non-Clifford Gates”, (2024) arXiv:2408.09254
Page edit log
- Victor V. Albert (2022-06-05) — most recent
- Victor V. Albert (2022-01-03)
- Ian Teixeira (2021-12-19)
Cite as:
“Quantum-double code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/quantum_double