## 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. This can equivalently constructed by braiding holes in a spherical geometry [2; Sec. 5].

## Protection

## Encoding

## Transversal Gates

## Gates

## Decoding

## Code Capacity Threshold

## Threshold

## Realizations

## Parent

- String-net code — The string-net model code for the category \(\text{Fib}\) is the Fibonacci string-net code.

## References

- [1]
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- [2]
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- [3]
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- [4]
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- [5]
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- [6]
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- [9]
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- [10]
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- [11]
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- [12]
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- [13]
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- [14]
- Y. Fan et al., “Experimental realization of a topologically protected Hadamard gate via braiding Fibonacci anyons”, (2022) arXiv:2210.12145
- [15]
- S. Xu et al., “Non-Abelian braiding of Fibonacci anyons with a superconducting processor”, Nature Physics (2024) arXiv:2404.00091 DOI
- [16]
- Z. K. Minev et al., “Realizing string-net condensation: Fibonacci anyon braiding for universal gates and sampling chromatic polynomials”, (2024) arXiv:2406.12820

## Page edit log

- Alexis Schotte (2022-01-24) — most recent
- Victor V. Albert (2022-01-24)

## Cite as:

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