Alternative names: Levin-Wen model code, Turaev-Viro code.
Description
A non-stabilizer commuting-projector code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a (multiplicity-free) unitary fusion category \( \mathcal{C} \). The code is defined on a cell decomposition dual to a triangulation of a two-dimensional surface, with a qudit of dimension \( |\mathcal{C}| \) located at each edge of the decomposition. These models realize local topological order (LTO) [5].
The codespace is the ground-state subspace of the Levin-Wen model commuting-projector Hamiltonian [1], a many-body Hamiltonian realizing the 3-manifold Turaev-Viro invariant [3,6]. This Hamiltonian is defined to act on a constrained Hilbert space defined by vertex constraints. Alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [3,7]. The fusion space can have dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes. Domain walls of string nets can be formulated using bimodule categories [8]. Anyon condensation of general string net models is studied in Ref. [9].
The initial formulation of string-net models [1] considered unitary fusion categories with an extra tetrahedral symmetry, but this was realized not to be necessary [2,3]. Explicit Hamiltonians for the more general categories have been studied [4].
Protection
Error-correcting properties established in Ref. [10].Encoding
For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [11] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [12,13].Scalable dynamic string-net preparation (DSNP) [14].String nets with solvable anyons and gappable boundaries [15; Def. 1.2] can be prepared via an adaptive finite-depth local unitary circuit [16].Gates
Gates can be implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles for triangulations of toroidal [17,18] and hyperbolic [19] manifolds. Whether or not a gate set is universal depends on the choice of input category; in some cases such as the Ising category, gates can be complemented by topological charge measurements to obtain a universal gate set.Alternatively, one could encode the logical quantum information into the degenerate fusion space of a number of computational anyons. In this case, a universal logical gate set can be implemented through the braiding of the computational anyons [3,7,20], e.g., for the case of the Fibonacci input category.Cousins
- Topological code— String-net codes realize 2D topological phases based on unitary fusion categories. Any 2D many-body state satisfying the entanglement bootstrap axioms can be mapped into the ground-state subspace of a string-net model via a constant-depth unitary circuit [23]. Different string-net models that have Morita-equivalent input fusion categories describing the same topological order are dual to each other [24].
- Cage-net code— The cage-net code is a non-stabilizer commuting-projector code obtained by coupling layers of \(G\)-graded string-net models [25].
- Walker-Wang model code— The Walker-Wang model is a generalization of the 3D version of the Levin-Wen model [1; Sec. 5], which realizes gauge theories coupled to bosons and fermions.
- X-cube model code— A non-stabilizer commuting-projector code constructed by stacking layers of the double-semion string-net model, called the semionic X-cube model [26], is equivalent to the X-cube model [27] (see also Refs. [28,29]).
Member of code lists
Primary Hierarchy
Parents
String-net code
Children
String-net models realize 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 [30].
String-net model ground states reduce to Hopf-algebra quantum-double ground states for categories \(\text{Rep}(H)\), where \(H\) is a Hopf algebra [31].
The string-net model code for the category \(\text{Fib}\) is the Fibonacci string-net code.
The string-net model code for the category \(\text{Vec}^{\omega}\mathbb{Z}_2\) for nontrivial cocycle is the double semion string-net code.
References
- [1]
- M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
- [2]
- S.-M. Hong, “On symmetrization of 6j-symbols and Levin-Wen Hamiltonian”, (2009) arXiv:0907.2204
- [3]
- R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010) arXiv:1002.2816 DOI
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- A. Hahn and R. Wolf, “Generalized string-net model for unitary fusion categories without tetrahedral symmetry”, Physical Review B 102, (2020) arXiv:2004.07045 DOI
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- C. Jones, P. Naaijkens, D. Penneys, and D. Wallick, “Local topological order and boundary algebras”, (2025) arXiv:2307.12552
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- M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation”, (2000) arXiv:quant-ph/0001108
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- [9]
- J. Christian, D. Green, P. Huston, and D. Penneys, “A lattice model for condensation in Levin-Wen systems”, (2023) arXiv:2303.04711
- [10]
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- [16]
- Y. Ren, N. Tantivasadakarn, and D. J. Williamson, “Efficient Preparation of Solvable Anyons with Adaptive Quantum Circuits”, (2024) arXiv:2411.04985
- [17]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits”, Physical Review Letters 125, (2020) arXiv:1806.02358 DOI
- [18]
- G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020) arXiv:1806.06078 DOI
- [19]
- A. Lavasani, G. Zhu, and M. Barkeshli, “Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes”, Quantum 3, 180 (2019) arXiv:1901.11029 DOI
- [20]
- M. H. Freedman, M. J. Larsen, and Z. Wang, “The Two-Eigenvalue Problem and Density¶of Jones Representation of Braid Groups”, Communications in Mathematical Physics 228, 177 (2002) arXiv:math/0103200 DOI
- [21]
- N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012) arXiv:1206.6048 DOI
- [22]
- G. Dauphinais and D. Poulin, “Fault-Tolerant Quantum Error Correction for non-Abelian Anyons”, Communications in Mathematical Physics 355, 519 (2017) arXiv:1607.02159 DOI
- [23]
- I. H. Kim and D. Ranard, “Classifying 2D topological phases: mapping ground states to string-nets”, (2024) arXiv:2405.17379
- [24]
- Y. Zhao and Y. Wan, “Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization”, (2025) arXiv:2408.02664
- [25]
- P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, “String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to \(p\)-String Condensation”, (2025) arXiv:2505.13604
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- H. Ma, E. Lake, X. Chen, and M. Hermele, “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
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- W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
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- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
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- S. Pai and M. Hermele, “Fracton fusion and statistics”, Physical Review B 100, (2019) arXiv:1903.11625 DOI
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- 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
- [31]
- B. Balsam and A. Kirillov Jr, “Kitaev’s Lattice Model and Turaev-Viro TQFTs”, (2012) arXiv:1206.2308
Page edit log
- Alexis Schotte (2022-01-24) — most recent
- David Aasen (2022-01-24)
- Victor V. Albert (2022-01-24)
Cite as:
“String-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/string_net