Haah cubic code (CC)[1] 


A 3D lattice stabilizer code on a length-\(L\) cubic lattice with one or two qubits per site. Admits two types of stabilizer generators with support on each cube of the lattice. In the non-CSS case, these two are related by spatial inversion. For CSS codes, we require that the product of all corner operators is the identity. We lastly require that there are no non-trival string operators, meaning that single-site operators are a phase, and any period one logical operator \(l \in \mathsf{S}^{\perp}\) is just a phase.

Haah showed in his original construction that there is exactly one non-CSS code of this form, and 17 CSS codes [1]. The non-CSS code is labeled code 0, and the rest are numbered from 1 - 17. Codes CC1-CC4, CC7, CC8, and CC10 do not have string logical operators [1,2].

Under renormalization group flow [3], cubic code 1 fragments into itself and the Haah B-code (a.k.a. CC1B), which has four qubits per unit cell [2; Eq. (D2)]. In this context, cubic code 1 is sometimes called the Haah A-code or CC1A.

Encodings using geometries with boundaries as well as lattice defects have been studied [4]. CC1A and CC1B have been generalized to manifolds more general than 3D lattices [5,6].


Cubic codes protect against simultaneous independent Pauli errors on different sites (not qubits, since there can be 2 qubits per site). Codes CC0-CC4 are known to have distance \(d \ge L\), meaning they can achieve macroscopic code distance as \(L\to\infty\).


Hard-decisions RG decoder [7].


The encoding rate depends on the code implemented, but code CC0 has been shown to have \(k \ge L\) (on a periodic finite cubic lattice of side length \(L\). In general we expect the number of logical bits to scale as \(k \sim L\).



  • 3D color code — The 3D color and cubic code families both include 3D codes that do not admit string-like operators.
  • Loop toric code — The energy of any partial implementation of CC1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
  • Generalized bicycle (GB) code — A GB code for the group \(G=\mathbb{Z}_3^{\times 3}\) is a cubic code [8; Sec. III.A].
  • Lifted-product (LP) code — A lifted-product code constructed with coefficients in the ring \(R=\mathbb{F}_2[x,y,z]/(x^L-1,y^L-1,z^L-1)\) is a cubic code [9; Appx. B].
  • Fibonacci code — The Fibonacci code is designed to mimic the fractal properties of (quantum) Haah cubic code so that studying the former can help us toward the development of an efficient algorithm for the latter [10].
  • Self-correcting quantum code — Cubic code 1 is partially self-correcting with a logarithmic energy barrier [7].


J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI
A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
J. Haah, “Bifurcation in entanglement renormalization group flow of a gapped spin model”, Physical Review B 89, (2014) arXiv:1310.4507 DOI
C. T. Aitchison et al., “Boundaries and defects in the cubic code”, Physical Review B 109, (2024) arXiv:2308.00138 DOI
K. T. Tian, E. Samperton, and Z. Wang, “Haah codes on general three-manifolds”, Annals of Physics 412, 168014 (2020) arXiv:1812.02101 DOI
K. T. Tian and Z. Wang, “Generalized Haah Codes and Fracton Models”, (2019) arXiv:1902.04543
S. Bravyi and J. Haah, “Quantum Self-Correction in the 3D Cubic Code Model”, Physical Review Letters 111, (2013) arXiv:1112.3252 DOI
P. Panteleev and G. Kalachev, “Quantum LDPC Codes With Almost Linear Minimum Distance”, IEEE Transactions on Information Theory 68, 213 (2022) arXiv:2012.04068 DOI
P. Panteleev and G. Kalachev, “Asymptotically Good Quantum and Locally Testable Classical LDPC Codes”, (2022) arXiv:2111.03654
G. M. Nixon and B. J. Brown, “Correcting Spanning Errors With a Fractal Code”, IEEE Transactions on Information Theory 67, 4504 (2021) arXiv:2002.11738 DOI
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Zoo Code ID: haah_cubic

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“Haah cubic code (CC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/haah_cubic
@incollection{eczoo_haah_cubic, title={Haah cubic code (CC)}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/haah_cubic} }
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“Haah cubic code (CC)”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/haah_cubic

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