# X-cube model code[1]

## Description

A foliated type-I fracton code supporting a subextensive number of logical qubits. Variants include the membrane-coupled [2], twice-foliated [3], and several generalized [4] X-cube models.

## Decoding

Parallelized matching decoder [5].

## Code Capacity Threshold

Independent \(X,Z\) noise: \(\approx 7.5\%\), higher than 3D surface code and color code [6].

## Parents

- Qubit CSS code
- Fracton stabilizer code — The X-cube model is a foliated type-I fracton code [7,8].

## Cousins

- Quantum-inspired classical block code — According to Ref. [9], a classical analogue of the X-cube model is the eight-vertex model [10–12].
- Newman-Moore code — Generalized X-cube models [4] are constructed from a product of the repetion (1D Ising) code and the Newman-Moore code.
- Quantum repetition code — Generalized X-cube models [4] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.
- Balanced product (BP) code — Generalized X-cube models [4] are constructed from a balanced product of the quantum repetion (1D Ising) code and the Newman-Moore code.
- Fracton Floquet code — The ISG of the X-cube Floquet code can be that of the X-cube model code or the checkerboard model code.
- X-cube Floquet code — The ISG of the X-cube Floquet code can be that of the X-cube model code or that of several decoupled surface codes.
- Majorana checkerboard code — The Majorana checkerboard code is equivalent via a constant-depth unitary to a semionic version of the X-cube model and some decoupled fermionic modes [13].
- Checkerboard model code — The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [14].

## References

- [1]
- 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
- [2]
- H. Ma et al., “Fracton topological order via coupled layers”, Physical Review B 95, (2017) arXiv:1701.00747 DOI
- [3]
- W. Shirley, K. Slagle, and X. Chen, “Fractional excitations in foliated fracton phases”, Annals of Physics 410, 167922 (2019) arXiv:1806.08625 DOI
- [4]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [5]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [6]
- H. Song et al., “Optimal Thresholds for Fracton Codes and Random Spin Models with Subsystem Symmetry”, Physical Review Letters 129, (2022) arXiv:2112.05122 DOI
- [7]
- W. Shirley, K. Slagle, and X. Chen, “Universal entanglement signatures of foliated fracton phases”, SciPost Physics 6, (2019) arXiv:1803.10426 DOI
- [8]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [9]
- G. M. Nixon and B. J. Brown, “Correcting Spanning Errors With a Fractal Code”, IEEE Transactions on Information Theory 67, 4504 (2021) arXiv:2002.11738 DOI
- [10]
- B. Sutherland, “Two-Dimensional Hydrogen Bonded Crystals without the Ice Rule”, Journal of Mathematical Physics 11, 3183 (1970) DOI
- [11]
- R. J. Baxter, “Eight-Vertex Model in Lattice Statistics”, Physical Review Letters 26, 832 (1971) DOI
- [12]
- R. J. Baxter, “Partition function of the Eight-Vertex lattice model”, Annals of Physics 70, 193 (1972) DOI
- [13]
- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
- [14]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI

## Page edit log

- Ke Liu (刘科 子竞) (2023-03-31) — most recent
- Victor V. Albert (2023-03-31)

## Cite as:

“X-cube model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/xcube