Fracton stabilizer code[1]
Description
A 3D translationally invariant modular-qudit stabilizer code whose codewords make up the ground-state space of a Hamiltonian in a fracton phase. Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted.
Qubit Fracton codes include the following three sub-types:
- (1)
Foliated type-I fracton phase: Excitations are mobile in less than 3 dimensions, but codes can be grown by foliation, i.e., stacking copies of the 2D surface code and applying a constant-depth circuit.
- (2)
Fractal type-I fracton phase: Excitations are mobile in less than 3 dimensions, and codes are not foliated.
- (3)
Type-II fracton phase: Excitations are not mobile in any dimension and there are no string operators.
Cousins
- Topological code— Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted.
- Kitaev surface code— Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [2; Eq. (D32)].
- Symmetry-protected topological (SPT) code— CSS fracton codes can be converted in 2D fractal-like SPT Hamiltonians [3].
- Groupoid toric code— Some groupiod toric code models admit fracton-like features such as extensive ground-state degeneracy and excitations with restricted mobility.
- Quantum repetition code— The 1D quantum repetition code is an ingredient in product constructions that yield several fracton phases [4; Fig. 8].
- XYZ color code— The XYZ color code resembles a Type-II fracton code in the limit of infinite noise bias [5].
- XZZX surface code— Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [6]. The XZZX surface code resembles a Type-I fracton code with lineons in the limit of infinite noise bias [5].
Primary Hierarchy
References
- [1]
- J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
- [2]
- 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
- [3]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [4]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [5]
- J. F. S. Miguel, D. J. Williamson, and B. J. Brown, “A cellular automaton decoder for a noise-bias tailored color code”, Quantum 7, 940 (2023) arXiv:2203.16534 DOI
- [6]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [7]
- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
- [8]
- 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
- [9]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI
- [10]
- D. J. Williamson and N. Baspin, “Layer Codes”, (2024) arXiv:2309.16503
- [11]
- B. Yoshida, “Exotic topological order in fractal spin liquids”, Physical Review B 88, (2013) arXiv:1302.6248 DOI
- [12]
- M. Pretko, X. Chen, and Y. You, “Fracton phases of matter”, International Journal of Modern Physics A 35, 2030003 (2020) arXiv:2001.01722 DOI
- [13]
- I. H. Kim, “3D local qupit quantum code without string logical operator”, (2012) arXiv:1202.0052
Page edit log
- Victor V. Albert (2022-01-05) — most recent
Cite as:
“Fracton stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fracton