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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 \((2,2)\) all-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

  • 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]. This process can be viewed as an ungauging [22–31] of certain symmetries. Several hybrid color-surface codes exist [32,33].
  • Quasi-hyperbolic color code— Quasi-hyperbolic color codes are related to quasi-hyperbolic surface codes via a constant-depth Clifford circuit [34].

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 [35].
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]
M. Levin and Z.-C. Gu, “Braiding statistics approach to symmetry-protected topological phases”, Physical Review B 86, (2012) arXiv:1202.3120 DOI
[23]
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
[24]
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
[25]
D. J. Williamson, “Fractal symmetries: Ungauging the cubic code”, Physical Review B 94, (2016) arXiv:1603.05182 DOI
[26]
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
[27]
A. Kubica and B. Yoshida, “Ungauging quantum error-correcting codes”, (2018) arXiv:1805.01836
[28]
W. Shirley, K. Slagle, and X. Chen, “Foliated fracton order from gauging subsystem symmetries”, SciPost Physics 6, (2019) arXiv:1806.08679 DOI
[29]
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
[30]
T. Rakovszky and V. Khemani, “The Physics of (good) LDPC Codes I. Gauging and dualities”, (2023) arXiv:2310.16032
[31]
D. J. Williamson and T. J. Yoder, “Low-overhead fault-tolerant quantum computation by gauging logical operators”, (2024) arXiv:2410.02213
[32]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[33]
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
[34]
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”, (2024) arXiv:2310.16982
[35]
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

Cite as:
“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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