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 stabilizer codes are commonly grouped into the following three sub-types [2]:
- (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 [3].
- (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.
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 [4,5].
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 [4,5].
- 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— Certain 3D CSS fracton codes can be ungauged [6–15] into 2D fractal-like SPT Hamiltonians; the paper gives an explicit construction from the 3D fractal code [11]. In subsystem-symmetry gauging [6–15] constructions, symmetry charges transforming under planar symmetries in one, two, or three directions become planon, lineon, or fracton excitations, respectively [12].
- Cage-net code— The cage-net construction can be used to realize various fracton phases, stabilizer and otherwise.
- Groupoid toric code— Some groupoid 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 [16; Fig. 8].
- Layer code— Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [17].
- XYZ color code— The XYZ color code resembles a Type-II fracton code in the limit of infinite noise bias [18].
- XZZX surface code— Subsystem symmetries play a role in finite-bias decoders for both XZZX and fracton codes [19]. The XZZX surface code resembles a Type-I fracton code with lineons in the limit of infinite noise bias [18].
Primary Hierarchy
Parents
Fracton stabilizer code
Children
The Majorana checkerboard code is a foliated type-I fracton code [20].
The checkerboard model is equivalent to two copies of the X-cube model via a local constant-depth unitary [22]. Hence, it is a foliated type-I fracton code.
The Fibonacci fractal spin-liquid code is a fractal type-I fracton code [2].
The Sierpinski prism model code is a fractal type-I fracton code [2].
Both HH-I and HH-II are fracton codes; HH-I is identified as foliated type-I, while HH-II remains inconclusive between fractal type-I and type-II in the sorting analysis of Ref. [2].
Both HHB models are expected to be foliated type-I fracton codes [2; Eqs. (D42-D43)].
The two-foliated fracton code is a foliated type-I fracton code.
The type-II fractal spin-liquid code is a type-II fracton code [23].
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 [2,24]. The qutrit models in [2; Eqs. (D11-D12)] are likely Type-II, with no string operators found numerically up to width 20, while the \(q=5\) qudit model in [2; Eq. (D13)] satisfies a proven no-string condition and is Type-II.
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]
- W. Shirley, K. Slagle, Z. Wang, and X. Chen, “Fracton Models on General Three-Dimensional Manifolds”, Physical Review X 8, (2018) arXiv:1712.05892 DOI
- [4]
- D. Aasen, D. Bulmash, A. Prem, K. Slagle, and D. J. Williamson, “Topological defect networks for fractons of all types”, Physical Review Research 2, (2020) arXiv:2002.05166 DOI
- [5]
- Z. Song, A. Dua, W. Shirley, and D. J. Williamson, “Topological Defect Network Representations of Fracton Stabilizer Codes”, PRX Quantum 4, (2023) arXiv:2112.14717 DOI
- [6]
- M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
- [7]
- J. Haegeman, K. Van Acoleyen, N. Schuch, J. I. Cirac, and F. Verstraete, “Gauging Quantum States: From Global to Local Symmetries in Many-Body Systems”, Physical Review X 5, (2015) arXiv:1407.1025 DOI
- [8]
- 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
- [9]
- D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
- [10]
- L. Bhardwaj, D. Gaiotto, and A. Kapustin, “State sum constructions of spin-TFTs and string net constructions of fermionic phases of matter”, Journal of High Energy Physics 2017, (2017) arXiv:1605.01640 DOI
- [11]
- A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
- [12]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
- [13]
- K. Dolev, V. Calvera, S. S. Cree, and D. J. Williamson, “Gauging the bulk: generalized gauging maps and holographic codes”, Journal of High Energy Physics 2022, (2022) arXiv:2108.11402 DOI
- [14]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
- [15]
- D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
- [16]
- T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
- [17]
- D. J. Williamson and N. Baspin, “Layer codes”, Nature Communications 15, (2024) arXiv:2309.16503 DOI
- [18]
- 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
- [19]
- B. J. Brown and D. J. Williamson, “Parallelized quantum error correction with fracton topological codes”, Physical Review Research 2, (2020) arXiv:1901.08061 DOI
- [20]
- T. Wang, W. Shirley, and X. Chen, “Foliated fracton order in the Majorana checkerboard model”, Physical Review B 100, (2019) arXiv:1904.01111 DOI
- [21]
- 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
- [22]
- W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order in the checkerboard model”, Physical Review B 99, (2019) arXiv:1806.08633 DOI
- [23]
- B. Yoshida, “Exotic topological order in fractal spin liquids”, Physical Review B 88, (2013) arXiv:1302.6248 DOI
- [24]
- M. Pretko, X. Chen, and Y. You, “Fracton phases of matter”, International Journal of Modern Physics A 35, 2030003 (2020) arXiv:2001.01722 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