# Modular-qudit surface code[1]

## Description

A family of stabilizer codes whose generators are few-body \(X\)-type and \(Z\)-type Pauli strings associated to the stars and plaquettes, respectively, of a tessellation of a two-dimensional surface (with a qudit located at each edge of the tesselation). The code has \( n=E \) many physical qudits, where \( E \) is the number of edges of the tesselation, and \( k=2g \) many logical qudits, where \( g \) is the genus of the surface.

## Protection

When defined on an \(L\times L\) square tiling of the torus, protects against \(L\) errors. More generally, the code distance is the number of edges in the shortest non contractible cycle in the tesselation or dual tesselation [2].

## Notes

The simplest Decodoku game is based on the qudit surface code with \( q=10\).

## Parents

- Modular-qudit CSS code — Plaquette and star operators are stabilizer generators.
- Abelian topological code — Qudit surface code Hamiltonians admit topological phases associated with \(\mathbb{Z}_q\) [3].

## Cousins

- Kitaev surface code — The qudit surface code with \(q=2\) is the surface code.
- String-net code — String-net model reduces to the qudit surface code when the category is the group \(\mathbb{Z}_q\).
- Quantum-double code — A quantum-double model with \(G=\mathbb{Z}_q\) is the qudit surface code.
- Translationally-invariant stabilizer code — Translation-invariant 2D prime-qudit topological stabilizer codes are equivalent to several copies of the prime-qudit surface code via a local constant-depth Clifford circuit [4].

## References

- [1]
- A. Y. Kitaev, “Fault-tolerant quantum computation by anyons”, Annals of Physics 303, 2 (2003). DOI; quant-ph/9707021
- [2]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002). DOI; quant-ph/0110143
- [3]
- S. S. Bullock and G. K. Brennen, “Qudit surface codes and gauge theory with finite cyclic groups”, Journal of Physics A: Mathematical and Theoretical 40, 3481 (2007). DOI; quant-ph/0609070
- [4]
- J. Haah, “Classification of translation invariant topological Pauli stabilizer codes for prime dimensional qudits on two-dimensional lattices”, Journal of Mathematical Physics 62, 012201 (2021). DOI; 1812.11193

## Page edit log

- Victor V. Albert (2022-01-05) — most recent
- Ian Teixeira (2021-12-19)

## Zoo code information

## Cite as:

“Modular-qudit surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/qudit_surface

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qudits/qudit_surface.yml.