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

## 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

- Projective-plane surface code
- Kitaev surface code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.
- 3D surface code
- Loop toric code
- 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
- Hypersphere product code
- Hyperbolic surface code

## 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 [13–15]. 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

- Victor V. Albert (2022-01-12) — most recent

## Cite as:

“Homological code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/higher_dimensional_surface