String-net code[1,2] 

Description

Also called a Turaev-Viro or Levin-Wen model code. Code whose codewords realize a 2D topological order rendered by a Turaev-Viro topological field theory. The corresponding anyon theory is defined by a 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.

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 [2,3]. Alternative constructions are possible, encoding information in the fusion space of the low-energy anyonic quasiparticle excitations of the model [2,4]. The fusion space can have dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.

String-net codes can equivalently be formulated in terms of unitary quantum groupoids [5].

Protection

Error-correcting properties established in Ref. [6].

Encoding

For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [7] or an \(O(\log{L})\)-depth unitary circuit with non-local two-qubit gates [8,9].

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 [10,11] and hyperbolic [12] 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 [2,4,13], e.g., for the case of the Fibonacci input category.

Decoding

Fusing non-Abelian anyons cannot be done in one step [14].Syndrome measurement circuits analyzed in Ref. [15].Clustering decoder [16].

Parents

  • Multi-fusion string-net code
  • Topological code — String-net codes can be realized using Levin-Wen model Hamiltonians, which realize 2D topological phases based on unitary fusion categories [13]. These models realize local topological order (LTO) [17].

Children

  • Fibonacci string-net code
  • Twisted quantum double (TQD) code — String-net models reduce to TQDs for categories \(\text{Vec}_G^\omega\), where \(G\) is a finite group, and \(\omega\) is a Type III cocycle. There is also a more direct duality between a large class of string–net models and certain TQD models [18].

Cousin

  • 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 with 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]
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010) arXiv:1002.2816 DOI
[3]
A. Kirillov Jr, “String-net model of Turaev-Viro invariants”, (2011) arXiv:1106.6033
[4]
M. Freedman, M. Larsen, and Z. Wang, “A modular functor which is universal for quantum computation”, (2000) arXiv:quant-ph/0001108
[5]
L. Chang, “Kitaev models based on unitary quantum groupoids”, Journal of Mathematical Physics 55, (2014) arXiv:1309.4181 DOI
[6]
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
[7]
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
[8]
M. Aguado and G. Vidal, “Entanglement Renormalization and Topological Order”, Physical Review Letters 100, (2008) arXiv:0712.0348 DOI
[9]
R. König, B. W. Reichardt, and G. Vidal, “Exact entanglement renormalization for string-net models”, Physical Review B 79, (2009) arXiv:0806.4583 DOI
[10]
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
[11]
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
[12]
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
[13]
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
[14]
D. Beckman et al., “Measurability of Wilson loop operators”, Physical Review D 65, (2002) arXiv:hep-th/0110205 DOI
[15]
N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012) arXiv:1206.6048 DOI
[16]
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
[17]
C. Jones et al., “Local topological order and boundary algebras”, (2023) arXiv:2307.12552
[18]
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
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Zoo Code ID: string_net

Cite as:
“String-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/string_net
BibTeX:
@incollection{eczoo_string_net, title={String-net code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/string_net} }
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“String-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/string_net

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/string_net.yml.