Fibonacci string-net code[1]

Description

Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.

The first type of encoding is into the ground-state subspace of the Levin-Wen model Hamiltonian, defined on a cell decomposition (dual to a triangulation) of a manifold with a qubit on each link. The code space is the simultaneous \(+1\) eigenspace of a set of vertex operators and plaquette operators, which are defined by the fusion rules and the numerical data of the Fibonacci category, respectively. The degeneracy of the code space is \(4g\), were \(g\) is the genus of the surface on which the cell decomposition is defined.

The second type of encoding is into the degenerate fusion space of a number of anyonic quasiparticle excitations of the Levin-Wen model.

Protection

When defined on a \(L \times L\) tailed honeycomb lattice on a torus, the code distance for ground-state encoding is \(L\).

Transversal Gates

A universal transversal gate set could be implemented in a folded version of this code using the techniques introduced in Ref.[2].

Gates

Universal gate set for the ground-state encoding is implemented through topological operations corresponding to elements of the mapping class group, which is generated by Dehn-twists along non-contractible cycles. These Dehn-twists can be implemented using constant-dept circuits when allowing long-range permutations of qubits [3][4].Universal gate set for the fusion-space encoding is implemented through braiding of the computational anyons [5][6].

Decoding

Clustering decoder (provides best known threshold for this code) [7].Fusion aware iterative minimum-weight perfect matching decoder. Note that ordinary MWPM decoders do not produce a threshold with this code [7].

Threshold

\(4.7\%\) for depolarizing noise, \(7.3\%\) for dephasing noise, and \(3.8\%\) for bit-flip noise with clustering decoder, assuming perfect measurements and gates [7].\(3.0\%\) for depolarizing noise, \(6.0\%\) for dephasing noise, and \(2.5\%\) for bit-flip noise with fusion-aware iterative MWPM decoder, assuming perfect measurements and gates [7].

Parent

Zoo code information

Internal code ID: fibonacci

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: fibonacci

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

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]
G. Zhu, M. Hafezi, and M. Barkeshli, “Quantum origami: Transversal gates for quantum computation and measurement of topological order”, Physical Review Research 2, (2020). DOI; 1711.05752
[3]
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
[4]
G. Zhu, A. Lavasani, and M. Barkeshli, “Instantaneous braids and Dehn twists in topologically ordered states”, Physical Review B 102, (2020). DOI; 1806.06078
[5]
Michael Freedman, Michael Larsen, and Zhenghan Wang, “A modular functor which is universal for quantum computation”. quant-ph/0001108
[6]
R. Koenig, G. Kuperberg, and B. W. Reichardt, “Quantum computation with Turaev–Viro codes”, Annals of Physics 325, 2707 (2010). DOI; 1002.2816
[7]
Alexis Schotte et al., “Quantum error correction thresholds for the universal Fibonacci Turaev-Viro code”. 2012.04610

Cite as:

“Fibonacci string-net code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fibonacci

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