# 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.

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.

Other fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects [2,3].

## Parent

## Children

- Majorana checkerboard code — The Majorana checkerboard code is a foliated type-I fracton code [4].
- Chamon model code — The Chamon model is a foliated type-I fracton code and is the first example of a fracton phase [5].
- Checkerboard model code — The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [6]. Hence, it is a foliated type-I fracton code.
- Fibonacci fractal spin-liquid code — The Fibonacci fractal spin-liquid code is a fractal type-I fracton code [5].
- Hsieh-Halasz (HH) code — Both HH-I and HH-II are fracton codes, with HH-I identified as a foliated type-I fracton code [5].
- Hsieh-Halasz-Balents (HHB) code — Both HHB models are expected to be foliated type-I fracton codes [5; Eqs. (D42-D43)].
- Layer code — Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [7].
- Sierpinsky fractal spin-liquid (SFSL) code — The Sierpinsky fractal spin-liquid code is a fractal type-I fracton code [5].
- X-cube model code — The X-cube model is a foliated type-I fracton code [5,8].
- Type-II fractal spin-liquid code — The type-II fractal spin-liquid code is a type-II fracton code [9].
- Qudit cubic code — Haah cubic [1] codes 1-4, 7, 8, and 10 do not have string logical operators and are the first examples of Type-II fracton phases. The remaining cubic codes are fractal Type-I fracton codes [5,10]. There is evidence that a qutrit and a \(q=5\) qudit cubic code from Ref. [11] have no string operators and are thus Type-II fracton codes (see [5; Eqs. (D11-D13)]).

## Cousins

- Topological code — Unlike topological phases, whose excitations can move in any direction, fracton phases are characterized by excitations whose movement is restricted. Fracton phases can be understood as topological defect networks, meaning that they can be described in the language of topological quantum field theory with defects [2,3].
- Kitaev surface code — Foliated type-I fracton phase codes can be grown by foliation, i.e., stacking copies of the 2D surface code; see [5; Eq. (D32)].
- Symmetry-protected topological (SPT) code — CSS fracton codes can be converted in 2D fractal-like SPT Hamiltonians [12].
- 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 [13; Fig. 8].
- XYZ color code — The XYZ color code resembles a Type-II fracton code in the limit of infinite noise bias [14].
- XZZX surface code — Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [15].

## References

- [1]
- J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
- [2]
- D. Aasen et al., “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
- [3]
- Z. Song et al., “Topological Defect Network Representations of Fracton Stabilizer Codes”, PRX Quantum 4, (2023) arXiv:2112.14717 DOI
- [4]
- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
- [5]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [6]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI
- [7]
- D. J. Williamson and N. Baspin, “Layer Codes”, (2024) arXiv:2309.16503
- [8]
- W. Shirley, K. Slagle, and X. Chen, “Universal entanglement signatures of foliated fracton phases”, SciPost Physics 6, (2019) arXiv:1803.10426 DOI
- [9]
- B. Yoshida, “Exotic topological order in fractal spin liquids”, Physical Review B 88, (2013) arXiv:1302.6248 DOI
- [10]
- M. Pretko, X. Chen, and Y. You, “Fracton phases of matter”, International Journal of Modern Physics A 35, 2030003 (2020) arXiv:2001.01722 DOI
- [11]
- I. H. Kim, “3D local qupit quantum code without string logical operator”, (2012) arXiv:1202.0052
- [12]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [13]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [14]
- 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
- [15]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI

## 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