Description
A foliated type-I fracton non-CSS code defined on a cubic lattice, with qubits on the even-parity sites and one weight-six stabilizer generator on each odd-parity site [3][2; Eq. (1)].
Each generator acts as \(X\), \(Y\), and \(Z\) on its two nearest-neighbor qubits along the \(x\), \(y\), and \(z\) axes, respectively [2; Eq. (1)]. A different choice of lattice vectors yields one qubit per vertex and one generator per cube, acting on six of its eight vertices [4; Eq. (D38)].
In the realization as an XYZ product of three repetition codes [5; Sec. 3.4], qubits live on the vertices and faces of a cubic lattice, and the natural logical operators are membrane-like rather than string-like [5].
Variants include a CSS model that is expected to have the same excitation structure [6] and a modified Chamon code based on the XYZ product code construction [5].
Protection
Flexible string operators confined to planes orthogonal to \([1,1,1]^T\) imply \(d = O(\sqrt{N})\) for the stabilizer version [5].Rate
For the stabilizer version on an \(n_1 \times n_2 \times n_3\) lattice, the number of logical qubits is \(k = 4 \gcd(n_1,n_2,n_3)\) [5].Decoding
Repetition-based decoder, based on the three underlying repetition codes and improved by pre-treatment with a probabilistic greedy local algorithm [7].Code Capacity Threshold
Depolarizing noise: \(4.92\%\) with repetition-based decoder [7].Cousins
- Repetition code— The Chamon model code can be obtained from an XYZ product of three repetition codes [8], in a construction different from the 3D surface code; see [5; Sec. 3.4].
- 3D surface code— The 3D planar and toric code on a cubic lattice can be obtained from a hypergraph product of three repetition codes [10][9; Exam. A.1]. The Chamon code is an XYZ product of three repetition codes [5; Sec. 3.4].
- XZZX surface code— The Chamon model code can be obtained from an XYZ product of three repetition codes [8]; see [5; Sec. 3.4]. Using only two repetition codes in the analogous 2D construction yields the XZZX code, making it a 2D analogue of the Chamon code [5; Sec. 2].
Primary Hierarchy
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]
- W. Shirley, X. Liu, and A. Dua, “Emergent fermionic gauge theory and foliated fracton order in the Chamon model”, Physical Review B 107, (2023) arXiv:2206.12791 DOI
- [4]
- A. Dua, I. H. Kim, M. Cheng, and D. J. Williamson, “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [5]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [6]
- 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
- [7]
- J. Zhao, Y.-C. Wu, and G.-P. Guo, “Quantum memory error correction computation based on Chamon model”, (2023) arXiv:2303.05267
- [8]
- D. Maurice, “Codes correcteurs quantiques pouvant se décoder itérativement”, PhD thesis, Université Pierre et Marie Curie-Paris VI, 2014.
- [9]
- L. Berent, T. Hillmann, J. Eisert, R. Wille, and J. Roffe, “Analog Information Decoding of Bosonic Quantum Low-Density Parity-Check Codes”, PRX Quantum 5, (2024) arXiv:2311.01328 DOI
- [10]
- W. Zeng and L. P. Pryadko, “Higher-Dimensional Quantum Hypergraph-Product Codes with Finite Rates”, Physical Review Letters 122, (2019) arXiv:1810.01519 DOI
Page edit log
- Victor V. Albert (2026-09-24) — most recent
- Victor V. Albert (2026-06-08)
- Zongyuan Wang (2025-04-18)
- Victor V. Albert (2025-04-18)
- Victor V. Albert (2024-02-08)
Cite as:
“Chamon model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/chamon, arXiv:2606.11484