# Freedman-Meyer-Luo code[1]

## Description

Hyperbolic surface code constructed using cellulation of a Riemannian Manifold \(M\) exhibiting 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.

The Freedman-Meter-Luo code has been generalized to a family with rate of order \(O(1/\sqrt{\log n})\) and minimum distance of order \(\Omega(\sqrt{\log n})\) which supports fault-tolerant non-Clifford gates [4].

## Protection

## Rate

## Fault Tolerance

## Notes

## Parent

## Cousin

- High-dimensional expander (HDX) code — Ramanujan codes broke 20-year record on minimum code distance set by Freedman-Meyer-Luo codes.

## References

- [1]
- “Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
- [2]
- M. H. Freedman, “Z\({}_{\text{2}}\)–Systolic-Freedom”, Proceedings of the Kirbyfest (1999) DOI
- [3]
- E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
- [4]
- G. Zhu et al., “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2023) arXiv:2310.16982
- [5]
- E. Fetaya, “Homological Error Correcting Codes and Systolic Geometry”, (2011) arXiv:1108.2886

## Page edit log

- Victor V. Albert (2022-01-03) — most recent
- Xinyuan Zheng (2021-12-18)

## Cite as:

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