Freedman-Meyer-Luo code[1]

Description

Hyperbolic surface code constructed using cellulation of a Riemann Manifold \(M\) exhibitng systolic freedom [2]. Codes derived from such manifolds can achieve distances scaling better than \(\sqrt{n}\), something that is impossible using closed 2D surfaces or 2D surfaces with boundaries [3]. Improved codes are obtained by studying a weak family of Riemann metrics on closed 4-dimensional manifolds \(S^2\otimes S^2\) with the \(Z_2\)-homology.

Protection

Four-dimensional manifolds with weak systolic freedom yield \([[n,2,\Omega(\sqrt{n \sqrt{\log n}})]]\) surface codes.

Rate

Codes held a 20-year record the best lower bound on asymptotic scaling of the minimum code distance, \(d=\Omega(\sqrt{n \sqrt{\log n}})\), broken by Ramanujan tensor-product codes.

Notes

See thesis by Fetaya for pedagogical exposition [4].

Parent

Cousin

Zoo code information

Internal code ID: freedman_meyer_luo

Your contribution is welcome!

on github.com (edit & pull request)

edit on this site

Zoo Code ID: freedman_meyer_luo

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

References

[1]
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002). DOI
[2]
M. H. Freedman, “Z2–Systolic-Freedom”, Proceedings of the Kirbyfest (1999). DOI
[3]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, 062202 (2012). DOI
[4]
Ethan Fetaya, “Homological Error Correcting Codes and Systolic Geometry”. 1108.2886

Cite as:

“Freedman-Meyer-Luo code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/freedman_meyer_luo

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/surface/freedman_meyer_luo.yml.