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][2] 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