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

Cousins

Zoo code information

Internal code ID: qudit_surface

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: qudit_surface

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

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

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.