3D surface code[1][2]


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.


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


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].



  • 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].


E. Dennis et al., “Topological quantum memory”, Journal of Mathematical Physics 43, 4452 (2002) arXiv:quant-ph/0110143 DOI
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
K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, (2023) arXiv:2212.06985
A. O. Quintavalle et al., “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
O. Higgott and N. P. Breuckmann, “Improved single-shot decoding of higher dimensional hypergraph product codes”, (2022) arXiv:2206.03122
A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
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
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
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Zoo Code ID: 3d_surface

Cite as:
“3D surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2022. https://errorcorrectionzoo.org/c/3d_surface
@incollection{eczoo_3d_surface, title={3D surface code}, booktitle={The Error Correction Zoo}, year={2022}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/3d_surface} }
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Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/tree/main/codes/quantum/qubits/topological/surface/3d_surface.yml.