Cubic theory code[1] 

Also known as Magic stabilizer code.


A geometrically local commuting-projector code defined on triangulations of lattices in arbitrary spatial dimensions. Its code Hamiltonian terms include Pauli-\(Z\) operators and products of Pauli-\(X\) operators and \(CZ\) gates. The Hamiltonian realizes higher-form \(\mathbb{Z}_2^3\) gauge theories whose excitations obey non-Abelian Ising-like fusion rules.

The construction is based on first constructing a model for an SPT phase [2,3], gauging its symmetries following Refs. [46], and making all terms commute outside of the ground-state subspace by projecting them to zero flux.


Probabilistic local cellular-automaton decoder [1].


  • Qubit code
  • Two-gauge theory code — Cubic theory codes realize higher-form \(\mathbb{Z}_2^3\) gauge theories with non-Abelian excitations in arbitrary dimensions.


  • Self-correcting quantum code — A family of five-dimensional cubic theory codes with non-Abelian excitations is argued to be self-correcting below a critical temperature via a Peierls argument [1].
  • Dihedral \(G=D_m\) quantum-double code — The ground-state subspace of the cubic theory model code in 2D reduces to that of the gauged \(G=\mathbb{Z}_3^2\) twisted quantum double model [1], realizing the same topological order as the \(G=D_4\) quantum double [7,8].


P.-S. Hsin, R. Kobayashi, and G. Zhu, “Non-Abelian Self-Correcting Quantum Memory”, (2024) arXiv:2405.11719
X. Chen et al., “Symmetry protected topological orders and the group cohomology of their symmetry group”, Physical Review B 87, (2013) arXiv:1106.4772 DOI
L. Tsui and X.-G. Wen, “Lattice models that realize Zn -1 symmetry-protected topological states for even n”, Physical Review B 101, (2020) arXiv:1908.02613 DOI
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
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
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
M. de W. Propitius, “Topological interactions in broken gauge theories”, (1995) arXiv:hep-th/9511195
L. Lootens et al., “Mapping between Morita-equivalent string-net states with a constant depth quantum circuit”, Physical Review B 105, (2022) arXiv:2112.12757 DOI
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Zoo Code ID: cubic_theory

Cite as:
“Cubic theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.
@incollection{eczoo_cubic_theory, title={Cubic theory code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={} }
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“Cubic theory code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024.