Homological code[14] 

Also known as Generalized surface code.

Description

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

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].

Decoding

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].

Notes

2D and 3D surface code visualization tool.

Parent

  • 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.

Children

Cousins

  • 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].

References

[1]
E. Dennis et al., “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]
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
[6]
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
[7]
E. Fetaya, “Bounding the distance of quantum surface codes”, Journal of Mathematical Physics 53, (2012) DOI
[8]
P. Webster and S. D. Bartlett, “Locality-preserving logical operators in topological stabilizer codes”, Physical Review A 97, (2018) arXiv:1709.00020 DOI
[9]
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
[10]
O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
[11]
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
[12]
D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
[13]
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
[14]
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
[15]
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
[16]
M. Vasmer and A. Kubica, “Morphing Quantum Codes”, PRX Quantum 3, (2022) arXiv:2112.01446 DOI
[17]
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
Page edit log

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

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
BibTeX:
@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} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/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

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