## Description

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]. 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

## Encoding

## Gates

## Decoding

## Parent

## Children

- Hopf-algebra quantum-double code — String-net model ground states reduce to Hopf-algebra quantum-double ground states for categories \(\text{Rep}(H)\), where \(H\) is a Hopf algebra [22].
- Fibonacci string-net code — The string-net model code for the category \(\text{Fib}\) is the Fibonacci string-net code.
- Twisted quantum double (TQD) code — 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 [23].

## 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 [24]. Different string-net models that have Morita-equivalent input fusion categories describing the same topological order are dual to each other [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.

## 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
- [4]
- 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
- [5]
- C. Jones et al., “Local topological order and boundary algebras”, (2023) arXiv:2307.12552
- [6]
- A. Kirillov Jr, “String-net model of Turaev-Viro invariants”, (2011) arXiv:1106.6033
- [7]
- M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation”, (2000) arXiv:quant-ph/0001108
- [8]
- L. Lootens et al., “Matrix product operator symmetries and intertwiners in string-nets with domain walls”, SciPost Physics 10, (2021) arXiv:2008.11187 DOI
- [9]
- J. Christian et al., “A lattice model for condensation in Levin-Wen systems”, (2023) arXiv:2303.04711
- [10]
- Y. Qiu and Z. Wang, “Ground subspaces of topological phases of matter as error correcting codes”, Annals of Physics 422, 168318 (2020) arXiv:2004.11982 DOI
- [11]
- Y.-J. Liu et al., “Methods for Simulating String-Net States and Anyons on a Digital Quantum Computer”, PRX Quantum 3, (2022) arXiv:2110.02020 DOI
- [12]
- M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
- [13]
- R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
- [14]
- Z. K. Minev et al., “Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials”, (2024) arXiv:2406.12820
- [15]
- 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
- [16]
- 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
- [17]
- 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
- [18]
- 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
- [19]
- D. Beckman et al., “Measurability of Wilson loop operators”, Physical Review D 65, (2002) arXiv:hep-th/0110205 DOI
- [20]
- N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012) arXiv:1206.6048 DOI
- [21]
- 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
- [22]
- B. Balsam and A. Kirillov Jr, “Kitaev’s Lattice Model and Turaev-Viro TQFTs”, (2012) arXiv:1206.2308
- [23]
- 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
- [24]
- I. H. Kim and D. Ranard, “Classifying 2D topological phases: mapping ground states to string-nets”, (2024) arXiv:2405.17379
- [25]
- Y. Zhao and Y. Wan, “Noninvertible Gauge Symmetry in (2+1)d Topological Orders: A String-Net Model Realization”, (2024) arXiv:2408.02664

## 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