Fracton stabilizer code

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].

Member of code lists

Primary Hierarchy

Quantum Domain

Modular-qudit Kingdom

Modular-qudit codeQECC Quantum

Modular-qudit USt code

Modular-qudit stabilizer codeStabilizer Hamiltonian-based QECC Quantum

3D lattice stabilizer codeLattice stabilizer QLDPC Stabilizer Hamiltonian-based QECC Quantum

Parents

3D lattice stabilizer code

Fracton stabilizer code

Children

Majorana checkerboard code

The Majorana checkerboard code is a foliated type-I fracton code [7].

Chamon model code

The Chamon model is a 4-foliated type-I fracton code [8] and is the first example of a fracton phase [2].

Checkerboard model code

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.

Fibonacci fractal spin-liquid code

The Fibonacci fractal spin-liquid code is a fractal type-I fracton code [2].

Hsieh-Halasz (HH) code

Both HH-I and HH-II are fracton codes, with HH-I identified as a foliated type-I fracton code [2].

Hsieh-Halasz-Balents (HHB) code

Both HHB models are expected to be foliated type-I fracton codes [2; Eqs. (D42-D43)].

Layer code

Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [10].

Sierpinsky fractal spin-liquid (SFSL) code

The Sierpinsky fractal spin-liquid code is a fractal type-I fracton code [2].

Two-foliated fracton code

The two-foliated fracton code is foliated type-I fracton code.

Type-II fractal spin-liquid code

The type-II fractal spin-liquid code is a type-II fracton code [11].

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 [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)]).

Qudit X-cube model code

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

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