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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

Parents
Fracton stabilizer code
Children
The Majorana checkerboard code is a foliated type-I fracton code [7].
The Chamon model is a 4-foliated type-I fracton code [8] and is the first example of a fracton phase [2].
The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [9]. Hence, it is a foliated type-I fracton code.
The Fibonacci fractal spin-liquid code is a fractal type-I fracton code [2].
Both HH-I and HH-II are fracton codes, with HH-I identified as a foliated type-I fracton code [2].
Both HHB models are expected to be foliated type-I fracton codes [2; Eqs. (D42-D43)].
Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [10].
The Sierpinsky fractal spin-liquid code is a fractal type-I fracton code [2].
The two-foliated fracton code is foliated type-I fracton code.
The type-II fractal spin-liquid code is a type-II fracton code [11].
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 [2,12]. There is evidence that a qutrit and a \(q=5\) qudit cubic code from Ref. [13] have no string operators and are thus Type-II fracton codes (see [2; Eqs. (D11-D13)]).

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
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Zoo Code ID: fracton

Cite as:
“Fracton stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fracton
BibTeX:
@incollection{eczoo_fracton, title={Fracton stabilizer code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/fracton} }
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“Fracton stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/fracton

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/fracton/fracton.yml.