## Description

A variant of the Kitaev surface code on a 3D lattice. The closely related solid code [3] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system.

## Decoding

Flip decoder and its modification p-flip [4].

## Threshold

Phenomenological noise model for the 3D toric code: \(2.90(2)\%\) under BP-OSD decoder [5], \(7.1\%\) under improved BP-OSD [6], and \(2.6\%\) under flip decoder [4]. For 3D surface code: \(3.08(4)\%\) under flip decoder [5].

## Parent

## Cousins

- Color code — Color code is equivalent to surface code in several ways [7][8]. For example, the color code on a \(D\)-dimensional closed manifold is equivalent to multiple decoupled copies of the \(D-1\)-dimensional surface code.
- Self-correcting quantum code — The 3D welded solid code is partially self-correcting with a power-law energy barrier [3]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [9].

## References

- [1]
- E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
- [2]
- A. Hamma, P. Zanardi, and X.-G. Wen, “String and membrane condensation on three-dimensional lattices”, Physical Review B 72, (2005) arXiv:cond-mat/0411752 DOI
- [3]
- K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
- [4]
- T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, (2023) arXiv:2212.06985
- [5]
- A. O. Quintavalle et al., “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
- [6]
- O. Higgott and N. P. Breuckmann, “Improved single-shot decoding of higher dimensional hypergraph product codes”, (2022) arXiv:2206.03122
- [7]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [8]
- 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
- [9]
- O. Landon-Cardinal et al., “Perturbative instability of quantum memory based on effective long-range interactions”, Physical Review A 91, (2015) arXiv:1501.04112 DOI

## Page edit log

- Aleksander Kubica (2022-05-16) — most recent
- Victor V. Albert (2022-05-16)

## Cite as:

“3D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/3d_surface