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Sierpinski prism model code[1,2]

Alternative names: Sierpinski fractal spin-liquid (SFSL) code, Yoshida first-order fractal spin-liquid code.

Description

A fractal type-I fracton CSS code defined on a cubic lattice [3; Eq. (D22)]. The code admits an excitation-moving operator shaped like a Sierpinski triangle [3; Fig. 2].

Cousins

  • Newman-Moore code— The Sierpinski prism model code is a hypergraph product of the repetition code and the Newman-Moore code [4,5].
  • Repetition code— The Sierpinski prism model code is a hypergraph product of the repetition code and the Newman-Moore code [4,5].
  • Symmetry-protected topological (SPT) code— Ungauging [6–15] yields explicit fracton-SPT constructions; in particular, the Yoshida first-order fractal spin-liquid code underlies a 2D fracton SPT protected by Sierpinski-triangle symmetries via a gapped domain wall [11].
  • 3D surface code— The Sierpinski prism model code admits a topological defect network construction out of 3D surface codes on triangular prisms [16,17].
  • Haah cubic code (CC)— The Haah A-code can be written in a similar form as the Sierpinski prism model code [17].

Member of code lists

Primary Hierarchy

Quantum Domain
Qubit Kingdom
Qubit codeQECC Quantum
Union stabilizer (USt) codeQECC Quantum
Qubit stabilizer codeStabilizer Hamiltonian-based Qubit QECC Quantum
Coherent-parity-check (CPC) code
Qubit CSS codeCSS Stabilizer Hamiltonian-based QECC Quantum
Generalized homological-product qubit CSS codeQLDPC Qubit Generalized homological-product CSS Stabilizer Hamiltonian-based QECC Quantum
Higher-dimensional homological product code
Hypergraph product (HGP) codeQLDPC CSS Generalized homological-product Lattice stabilizer Stabilizer Hamiltonian-based Qubit QECC Quantum
Parents
Hypergraph product (HGP) code
The Sierpinski prism model code is a hypergraph product of the repetition code and the Newman-Moore code [4,5].
Fracton stabilizer codeLattice stabilizer Stabilizer Hamiltonian-based QECC Quantum
The Sierpinski prism model code is a fractal type-I fracton code [3].
Sierpinski prism model code

References

[1]
C. Castelnovo and C. Chamon, “Topological quantum glassiness”, Philosophical Magazine 92, 304 (2012) arXiv:1108.2051 DOI
[2]
B. Yoshida, “Exotic topological order in fractal spin liquids”, Physical Review B 88, (2013) arXiv:1302.6248 DOI
[3]
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
[4]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes II. Product constructions”, (2024) arXiv:2402.16831
[5]
Y. Tan, B. Roberts, N. Tantivasadakarn, B. Yoshida, and N. Y. Yao, “Fracton models from product codes”, Physical Review Research 7, (2025) arXiv:2312.08462 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]
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
[17]
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
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Zoo Code ID: sierpinsky_fractal_liquid

Cite as:
“Sierpinski prism model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/sierpinsky_fractal_liquid
BibTeX:
@incollection{eczoo_sierpinsky_fractal_liquid, title={Sierpinski prism model code}, booktitle={The Error Correction Zoo}, year={2026}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/sierpinsky_fractal_liquid} }
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Cite as:

“Sierpinski prism model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/sierpinsky_fractal_liquid

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