3D surface code

3D surface code[1,2]

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].

Parents

Cousins

Single-shot code — Some 3D surface codes are single-shot codes [7].
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 [8], but it is not a quantum self-correcting memory [9].
3D subsystem surface code

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”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
[7]: C. Stahl, “Single-Shot Quantum Error Correction in Intertwined Toric Codes”, (2023) arXiv:2307.08118
[8]: 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
[9]: Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389

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

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