Welcome to the Category Kingdom.

Category-based quantum code
Encodes a finite-dimensional logical Hilbert space into a physical Hilbert space associated with a category. Often associated with a particular topological quantum field theory (TQFT), as the data of such theories is described by a category.
Parents:
Finite-dimensional quantum error-correcting code.
Parent of:
String-net code.

String-net code[1][2]
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).
Protection: Error-correcting properties established in Ref. [3].
Parents:
Category-based quantum code, Topological code.
Parent of:
Fibonacci string-net code.
Cousin of:
Kitaev surface code, Modular-qudit surface code, Quantum-double code.

Fibonacci string-net code[1]
Quantum error correcting code associated with the Levin-Wen string-net model with the Fibonacci input category, admitting two types of encodings.
Protection: When defined on a \(L \times L\) tailed honeycomb lattice on a torus, the code distance for ground-state encoding is \(L\).
Parents:
String-net code.

## 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]
- 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