Homological code[14] 

Also known as Generalized surface code.


CSS-type extenstion of the Kitaev surface code to arbitrary manifolds. The version on a Euclidean manifold of some fixed dimension is called the \(D\)-dimensional surface or \(D\)-dimensional toric code.

Given a cellulation of a manifold, qubits are put on \(i\)-dimensional faces, \(X\)-type stabilizers are associated with \((i-1)\)-faces, while \(Z\)-type stabilizers are associated with \((i+1)\)-faces.

The 4D surface code serves as a self-correcting quantum memory, while surface codes in higher dimensions can have distances not possible in lower dimensions.


Rate depends on the underlying cellulation and manifold [1,4]. For general 2D manifolds, \(kd^2\leq c(\log k)^2 n\) for some constant \(c\) [5], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [6,7], and (2) surface codes with finite rate can only achieve an asymptotic minimum distance that is logarithmic in \(n\). Higher-dimensional manifolds yield distances scaling more favorably. Loewner's theorem provides an upper bound for any bounded-geometry surface code [2].

Transversal Gates

Locality preserving operations can be determined for stacks of homoogical codes in any dimension [8].


Sweep decoder [9].Improved BP-OSD decoder [10].Renormalization-group (RG) decoder [11].

Code Capacity Threshold

\(>0\%\) threshold with sweep decoder for lattice surface codes in various dimensions [9].


2D and 3D surface code visualization tool.


  • Generalized homological-product qubit CSS code — The generalized surface code is constructed from chain complexes arising from cell complexes of the underlying manifold. Such complexes are not necessarily products of two non-trivial complexes, but the manifolds are picked so that their homology ensures favorable code properties.



  • Cycle code — Cycle codes feature in generalizations of the surface code [3].
  • \([[2^D,D,2]]\) hypercube code — The hypercube code can be concatenated with a distance-two \(D\)-dimensional surface code to yield a \([[2^D(2^D+1),D,4]]\) error-correcting code family that admits a transversal implementation of the logical \(C^{D-1}Z\) gate [12].
  • Color code — The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D\)-dimensional surface code via a local constant-depth Clifford circuit [1315]. Several hybrid color-surface codes exist [16,17].


E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
G. Zémor, “On Cayley Graphs, Surface Codes, and the Limits of Homological Coding for Quantum Error Correction”, Lecture Notes in Computer Science 259 (2009) DOI
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory (2013) arXiv:1301.6588 DOI
S. Bravyi, D. Poulin, and B. Terhal, “Tradeoffs for Reliable Quantum Information Storage in 2D Systems”, Physical Review Letters 104, (2010) arXiv:0909.5200 DOI
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
P. Webster and S. D. Bartlett, “Locality-preserving logical operators in topological stabilizer codes”, Physical Review A 97, (2018) arXiv:1709.00020 DOI
A. M. Kubica, The ABCs of the Color Code: A Study of Topological Quantum Codes as Toy Models for Fault-Tolerant Quantum Computation and Quantum Phases Of Matter, California Institute of Technology, 2018 DOI
O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
K. Duivenvoorden, N. P. Breuckmann, and B. M. Terhal, “Renormalization Group Decoder for a Four-Dimensional Toric Code”, IEEE Transactions on Information Theory 65, 2545 (2019) arXiv:1708.09286 DOI
D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
B. Yoshida, “Classification of quantum phases and topology of logical operators in an exactly solved model of quantum codes”, Annals of Physics 326, 15 (2011) arXiv:1007.4601 DOI
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
A. B. Aloshious, A. N. Bhagoji, and P. K. Sarvepalli, “On the Local Equivalence of 2D Color Codes and Surface Codes with Applications”, (2018) arXiv:1804.00866
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
N. Shutty and C. Chamberland, “Decoding Merged Color-Surface Codes and Finding Fault-Tolerant Clifford Circuits Using Solvers for Satisfiability Modulo Theories”, Physical Review Applied 18, (2022) arXiv:2201.12450 DOI
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Zoo Code ID: higher_dimensional_surface

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_dimensional_surface.yml.