Also known as Chamon-Bravyi-Leemhuis-Terhal (CBLT) code.

## Description

A foliated type-I fracton non-CSS code defined on a cubic lattice using one weight-eight stabilizer generator acting on the eight vertices of each cube in the lattice [3; Eq. (D38)].

Variants include a CSS model that is expected to have the same excitation structure [4] and a modified Chamon code based on the XYZ product code construction [5].

## Rate

The number of logical qubits is \(k = \text{gcd}(a,b,c)\) for a code constructed as a hypergraph product of three repetition codes of length \(a\), \(b\), and \(c\), respectively [2].

## Decoding

Repetition-based decoder, based on the three underlying repetition codes and improved by pre-treatment with a probabilistic greedy local algorithm [6].

## Code Capacity Threshold

Depolarizing noise: \(4.92\%\) with repetition-based decoder [6].

## Parents

- Fracton stabilizer code — The Chamon model is a foliated type-I fracton code and is the first example of a fracton phase [3].
- XYZ product code — The Chamon model code can be obtained from a hypergraph product of three repetition codes [7], but done in a different way than the 3D surface code; see [5; Sec. 3.4].

## Cousins

- 3D surface code — The Chamon and 3D surface codes can both be built out of a hypergraph product of three repetition codes; see [5; Sec. 3.4].
- XZZX surface code — The Chamon model code can be obtained from a particular hypergraph product of three repetition codes [7]; see [5; Sec. 3.4]. Using only two repetition codes yields the XZZX code, making that code a 2D analogue of the Chamon code [5; Sec. 2].

## References

- [1]
- C. Chamon, “Quantum Glassiness in Strongly Correlated Clean Systems: An Example of Topological Overprotection”, Physical Review Letters 94, (2005) arXiv:cond-mat/0404182 DOI
- [2]
- S. Bravyi, B. Leemhuis, and B. M. Terhal, “Topological order in an exactly solvable 3D spin model”, Annals of Physics 326, 839 (2011) arXiv:1006.4871 DOI
- [3]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [4]
- S. Vijay, J. Haah, and L. Fu, “Fracton topological order, generalized lattice gauge theory, and duality”, Physical Review B 94, (2016) arXiv:1603.04442 DOI
- [5]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [6]
- J. Zhao, Y.-C. Wu, and G.-P. Guo, “Quantum memory error correction computation based on Chamon model”, (2023) arXiv:2303.05267
- [7]
- Maurice, Denise. Codes correcteurs quantiques pouvant se décoder itérativement. Diss. Université Pierre et Marie Curie-Paris VI, 2014.

## Page edit log

- Victor V. Albert (2024-02-08) — most recent

## Cite as:

“Chamon model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/chamon