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Homological code[1–4]

Alternative names: Generalized surface code.

Description

CSS-type extension 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.

Lattice surface codes in \(D\) spatial dimensions can be partially classified by the dimension of their stabilizer generators (and corresponding excitations). There are \((p,q)\) surface codes for \(p+q=D\) realized by \(Z\)-type stabilizer generators of dimension \(p\) and \(X\)-type stabilizer generators of dimension \(q\). The two corresponding types of excitations are of dimension \(p-1\) and \(q-1\), respectively.

In addition, one has to classify the self statistics of the codes' excitations. For example, there are three types of \((1,3)\) surface codes in 3D, corresponding to the three types of \(\mathbb{Z}_2\) Abelian topological orders: one with bosonic charge and loop excitations (BcBl) and two with fermionic charge excitations and bosonic (FcBl) and fermionic (FcFl) loop excitations, respectively [5,6]. There exists an invariant that distinguishes these [6].

Protection

The 4D \((2,2)\) loop surface code serves as a self-correcting quantum memory, while surface codes in higher dimensions can have distances not possible in lower dimensions.

Rate

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 \(c\) [7], meaning that (1) 2D surface codes with bounded geometry have distance scaling at most as \(O(\sqrt{n})\) [8,9], 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 [10].

Gates

Lattice surgery exists for 3D and 4D surface codes [11].

Decoding

Local automaton decoders based on Toom's rule and its generalization, the sweep rule [12–14].Improved BP-OSD decoder [15].Renormalization group (RG) decoder [16].

Code Capacity Threshold

\(>0\%\) threshold with sweep decoder for lattice surface codes in various dimensions [14].

Notes

2D and 3D surface code visualization tool. [17] on the role of homology in constructing surface codes by D. Browne.

Cousins

  • Lattice stabilizer code— Lattice surface codes in \(D\) spatial dimensions can be partially classified by the dimension of their stabilizer generators (and corresponding excitations). There are \((p,q)\) surface codes for \(p+q=D\) realized by \(Z\)-type stabilizer generators of dimension \(p\) and \(X\)-type stabilizer generators of dimension \(q\). The two corresponding types of excitations are of dimension \(p-1\) and \(q-1\), respectively.
  • Cycle code— Cycle codes feature in generalizations of the surface code [3].
  • \([[2^D,D,2]]\) hypercube quantum code— The hypercube quantum 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 [18].
  • 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 [19–21] (see also [22; Exam. 4]). The reverse of this process can be viewed as gauging [23–32] certain symmetries. Several hybrid color-surface codes exist [33,34].
  • Quasi-hyperbolic color code— Quasi-hyperbolic color codes are related to quasi-hyperbolic surface codes via a constant-depth Clifford circuit [35].

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 codeGeneralized homological-product QLDPC CSS Stabilizer Hamiltonian-based QECC Quantum
Parents
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.
Homological code
Children
Kitaev surface code
The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
3D surface code
\((1,3)\) 4D toric code
The \((1,3)\) 4D toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with 1D \(Z\)-type and 3D \(X\)-type logical operators.
Loop toric code
The 4D loop toric code realizes 4D \(\mathbb{Z}_2\) gauge theory with only loop excitations [36].
Fractal surface code
Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
Hemicubic code
\(D\)-dimensional twisted toric code
Hurwitz surface code
Hypersphere product code
Hyperbolic surface code

References

[1]
E. Dennis, A. Kitaev, A. Landahl, and J. Preskill, “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
[2]
“Z2-systolic freedom and quantum codes”, Mathematics of Quantum Computation 303 (2002) DOI
[3]
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
[4]
N. Delfosse, P. Iyer, and D. Poulin, “Generalized surface codes and packing of logical qubits”, (2016) arXiv:1606.07116
[5]
T. Johnson-Freyd, “(3+1)D topological orders with only a \(\mathbb{Z}_2\)-charged particle”, (2020) arXiv:2011.11165
[6]
L. Fidkowski, J. Haah, and M. B. Hastings, “Gravitational anomaly of (3+1) -dimensional Z2 toric code with fermionic charges and fermionic loop self-statistics”, Physical Review B 106, (2022) arXiv:2110.14654 DOI
[7]
N. Delfosse, “Tradeoffs for reliable quantum information storage in surface codes and color codes”, 2013 IEEE International Symposium on Information Theory 917 (2013) arXiv:1301.6588 DOI
[8]
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
[9]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
[10]
P. Webster and S. D. Bartlett, “Locality-preserving logical operators in topological stabilizer codes”, Physical Review A 97, (2018) arXiv:1709.00020 DOI
[11]
T. Hillmann, G. Dauphinais, I. Tzitrin, and M. Vasmer, “Single-shot and measurement-based quantum error correction via fault complexes”, (2024) arXiv:2410.12963
[12]
C. S. Ahn, Extending Quantum Error Correction: New Continuous Measurement Protocols and Improved Fault-Tolerant Overhead, California Institute of Technology, 2004 DOI
[13]
A. Kubica and J. Preskill, “Cellular-Automaton Decoders with Provable Thresholds for Topological Codes”, Physical Review Letters 123, (2019) arXiv:1809.10145 DOI
[14]
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
[15]
O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
[16]
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
[17]
D. Browne, Lecture notes
[18]
D. Hangleiter, M. Kalinowski, D. Bluvstein, M. Cain, N. Maskara, X. Gao, A. Kubica, M. D. Lukin, and M. J. Gullans, “Fault-Tolerant Compiling of Classically Hard Instantaneous Quantum Polynomial Circuits on Hypercubes”, PRX Quantum 6, (2025) arXiv:2404.19005 DOI
[19]
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
[20]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[21]
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
[22]
X. Fu, H. Zheng, Z. Li, and Z.-W. Liu, “No-go theorems for logical gates on product quantum codes”, (2025) arXiv:2507.16797
[23]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[24]
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
[25]
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
[26]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[27]
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
[28]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[29]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[30]
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
[31]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[32]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[33]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[34]
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
[35]
G. Zhu, S. Sikander, E. Portnoy, A. W. Cross, and B. J. Brown, “Non-Clifford and parallelizable fault-tolerant logical gates on constant and almost-constant rate homological quantum LDPC codes via higher symmetries”, (2025) arXiv:2310.16982
[36]
X. Chen, A. Dua, P.-S. Hsin, C.-M. Jian, W. Shirley, and C. Xu, “Loops in 4+1d topological phases”, SciPost Physics 15, (2023) arXiv:2112.02137 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
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@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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