Plaquette Ising code

Plaquette Ising code[1–3]

Alternative names: Right-angle water ice code, Xu-Moore code.

Description

Classical code defined on a cubic lattice in usually two or three dimensions whose parity checks are applied on the four vertices of each square.

Protection

The 2D code has parameters \([L^2,2L-1,L]\) on a square lattice of size \(L\) [4]. The 3D version has been studied in Ref. [5].

Cousins

Rotated surface code— The 2D plaquette Ising model can be thought of as the rotated surface code whose \(X\)-type stabilizer generators have been converted to \(Z\)-type stabilizer generators.
X-cube model code— The 3D plaquette Ising model can be used to obtain the X-cube model by gauging [6–15] its subsystem symmetry [8].
Abelian topological code— The 2D plaquette Ising model can be constructed by coupling layers of 1D \(\mathbb{Z}_2\) lattice gauge theory [16]. A field-theoretic description of the 2D plaquette Ising model can be obtained by coupling layers of 1D gauge theory [17].
Two-foliated fracton code— The two-foliated fracton code is a hypergraph product of the repetition code and the plaquette Ising code on a square lattice with periodic boundary conditions [4].

Member of code lists

Primary Hierarchy

Classical Domain

Binary Kingdom

Binary codeECC

Binary group-orbit codeECC

Linear binary codeLinear \(q\)-ary Linear code over \(\mathbb{Z}_q\) ECC

Classical topological codeQuantum-inspired classical block ECC

Parents

Classical topological code

The 2D plaquette Ising model can be constructed by coupling layers of 1D \(\mathbb{Z}_2\) lattice gauge theory [16]. A field-theoretic description of the 2D plaquette Ising model can be obtained by coupling layers of 1D gauge theory [17].

Plaquette Ising code

References

[1]: Ziman, John M. Models of disorder: the theoretical physics of homogeneously disordered systems. Cambridge university press, 1979.
[2]: J. E. Moore and D.-H. Lee, “Geometric effects onT-breaking inp+ipandd+idsuperconducting arrays”, Physical Review B 69, (2004) arXiv:cond-mat/0309717 DOI
[3]: C. Xu and J. E. Moore, “Strong-Weak Coupling Self-Duality in the Two-Dimensional Quantum Phase Transition ofp+ipSuperconducting Arrays”, Physical Review Letters 93, (2004) arXiv:cond-mat/0312587 DOI
[4]: N. P. Breuckmann, M. Davydova, J. N. Eberhardt, and N. Tantivasadakarn, “Cups and Gates I: Cohomology invariants and logical quantum operations”, (2024) arXiv:2410.16250
[5]: D. A. Johnston, M. Mueller, and W. Janke, “Plaquette Ising models, degeneracy and scaling”, The European Physical Journal Special Topics 226, 749 (2017) arXiv:1612.00060 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]: B. Rayhaun and D. Williamson, “Higher-form subsystem symmetry breaking: Subdimensional criticality and fracton phase transitions”, SciPost Physics 15, (2023) arXiv:2112.12735 DOI
[17]: P. Gorantla, A. Prem, N. Tantivasadakarn, and D. J. Williamson, “String-Membrane-Nets from Higher-Form Gauging: An Alternate Route to \(p\)-String Condensation”, (2025) arXiv:2505.13604

Page edit log

Victor V. Albert (2025-01-16) — most recent

Cite as:

“Plaquette Ising code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2025. https://errorcorrectionzoo.org/c/plaquette_ising

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/classical/bits/quantum_inspired/plaquette_ising.yml.