String-net code

String-net code[1][2]

Description

Also called a Turaev-Viro or Levin-Wen model code. A family of topological codes, defined by a finite unitary spherical category \( \mathcal{C} \), whose generators are few-body operators acting 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 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 [4][2]. The fusion space can have dimension greater than one, allowing for topological quantum computation of logical information stored in the fusion outcomes.

Protection

Error-correcting properties established in Ref. [5].

Encoding

For an \(L\times L\) lattice, deterministic state preparation can be done with a geometrically local unitary \(O(L)\)-depth circuit [6].

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 [7][8] and hyperbolic [9] 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 [4][10][2], e.g., for the case of the Fibonacci input category.

Decoding

Syndrome measurement circuits analyzed in Ref. [11].

Parents

Category-based quantum code
Topological code — String-net codes can be realized using Levin-Wen model Hamiltonians, which realize various topological phases [1][2][3].

Child

Fibonacci string-net code

Cousins

Kitaev surface code — String-net model reduces to the surface code when the category is the group \(\mathbb{Z}_2\).
Modular-qudit surface code — String-net model reduces to the qudit surface code when the category is the group \(\mathbb{Z}_q\).
Quantum-double code — String-net model reduces to the quantum-double model for group categories.

References

[1]: M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005). DOI; cond-mat/0404617
[2]: R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
[3]: Alexander Kirillov Jr, “String-net model of Turaev-Viro invariants”. 1106.6033
[4]: Michael Freedman, Michael Larsen, and Zhenghan Wang, “A modular functor which is universal for quantum computation”. quant-ph/0001108
[5]: Y. Qiu and Z. Wang, “Ground subspaces of topological phases of matter as error correcting codes”, Annals of Physics 422, 168318 (2020). DOI; 2004.11982
[6]: Yu-Jie Liu et al., “Methods for simulating string-net states and anyons on a digital quantum computer”. 2110.02020
[7]: G. Zhu, A. Lavasani, and M. Barkeshli, “Universal Logical Gates on Topologically Encoded Qubits via Constant-Depth Unitary Circuits”, Physical Review Letters 125, (2020). DOI; 1806.02358
[8]: G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
[9]: A. Lavasani, G. Zhu, and M. Barkeshli, “Universal logical gates with constant overhead: instantaneous Dehn twists for hyperbolic quantum codes”, Quantum 3, 180 (2019). DOI; 1901.11029
[10]: 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). DOI; math/0103200
[11]: N. E. Bonesteel and D. P. DiVincenzo, “Quantum circuits for measuring Levin-Wen operators”, Physical Review B 86, (2012). DOI; 1206.6048

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

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