## 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].

## Decoding

Improved BP-OSD decoder [8].

## Threshold

Phenomenological noise model for the 4D toric code: \(4.3\%\) under improved BP-OSD decoder [8].

## 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

- 3D fermionic surface code
- 3D surface code
- Fractal surface code — Fractal surface codes are obtained by removing qubits from the surface code on a cubic lattice.
- Hemicubic code
- Hypersphere product code
- Hyperbolic surface code
- Projective-plane surface code
- Kitaev surface code — The surface-code CSS stabilizer generator prescription is extendable to higher-dimensional manifolds.

## Cousins

- Self-correcting quantum code — The 4D toric code is a self-correcting quantum memory [1,9].
- Haah cubic code — The energy of any partial implementation of Haah cubic code 1 is proportional to the boundary length similar to the 4D toric code, which can potentially surpress the effects of thermal errors, but it is currently an open problem.
- Color code — The color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D\)-dimensional surface code [10–12].

## 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]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [9]
- R. Alicki et al., “On thermal stability of topological qubit in Kitaev’s 4D model”, (2008) arXiv:0811.0033
- [10]
- 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
- [11]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [12]
- 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

## Page edit log

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

## Cite as:

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