## Description

A generalization of the Kitaev surface code defined on a 3D lattice.

3D toric code often either refers to the construction on the three-dimensional torus or is an alternative name for the general construction. The construction on surfaces with boundaries is often called the 3D surface code. There exists a rotated version of the 3D surface code, the 3D rotated surface code, akin to the (2D) rotated surface code [3]. The welded surface code [4] consists of several 3D surface codes stitched together in a way that the distance scales faster than the linear size of the system.

Related models [5,6] on lattices with certain colorability are equivalent to several copies of the 3D surface code [7].

## Protection

## Transversal Gates

## Gates

## Decoding

## Fault Tolerance

## Code Capacity Threshold

## Threshold

## Parents

- Homological code
- 3D lattice stabilizer code
- Abelian topological code — The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory.
- XYZ product code — The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [18; Exam. A.1], but done in a different way than the Chamon code; see [19; Sec. 3.4].

## Cousins

- Chamon model code — The Chamon and 3D surface codes can both be built out of a hypergraph product of three repetition codes; see [19; Sec. 3.4].
- Rotated surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [3].
- Self-correcting quantum code — The 3D welded surface code is partially self-correcting with a power-law energy barrier [4]. The 3D toric code is a classical self-correcting memory, whose protected bit admits a membrane-like logical operator [20], but it is not a quantum self-correcting memory [21].
- 3D color code — The 3D color code is equivalent to multiple decoupled copies of the 3D surface code via a local constant-depth Clifford circuit [8,22,23]. This mapping can also be done via code concatenation [9].
- 3D fermionic surface code — The 3D (fermionic) surface code is a CSS (non-CSS) code which realizes a \(\mathbb{Z}_2\) gauge theory in 3D (with an emergent fermion).
- 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]
- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
- [4]
- K. P. Michnicki, “3D Topological Quantum Memory with a Power-Law Energy Barrier”, Physical Review Letters 113, (2014) arXiv:1406.4227 DOI
- [5]
- H. Bombin and M. A. Martin-Delgado, “Exact topological quantum order inD=3and beyond: Branyons and brane-net condensates”, Physical Review B 75, (2007) arXiv:cond-mat/0607736 DOI
- [6]
- I. H. Kim, “Local non-CSS quantum error correcting code on a three-dimensional lattice”, (2013) arXiv:1012.0859
- [7]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [8]
- A. Kubica, B. Yoshida, and F. Pastawski, “Unfolding the color code”, New Journal of Physics 17, 083026 (2015) arXiv:1503.02065 DOI
- [9]
- M. Vasmer and D. E. Browne, “Three-dimensional surface codes: Transversal gates and fault-tolerant architectures”, Physical Review A 100, (2019) arXiv:1801.04255 DOI
- [10]
- M. Barkeshli et al., “Higher-group symmetry in finite gauge theory and stabilizer codes”, (2024) arXiv:2211.11764
- [11]
- T. R. Scruby and K. Nemoto, “Local Probabilistic Decoding of a Quantum Code”, Quantum 7, 1093 (2023) arXiv:2212.06985 DOI
- [12]
- C. Piveteau, C. T. Chubb, and J. M. Renes, “Tensor Network Decoding Beyond 2D”, (2023) arXiv:2310.10722
- [13]
- A. Kulkarni and P. K. Sarvepalli, “Decoding the three-dimensional toric codes and welded codes on cubic lattices”, Physical Review A 100, (2019) arXiv:1808.03092 DOI
- [14]
- N. Delfosse and G. Zémor, “Linear-time maximum likelihood decoding of surface codes over the quantum erasure channel”, Physical Review Research 2, (2020) arXiv:1703.01517 DOI
- [15]
- 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
- [16]
- A. O. Quintavalle et al., “Single-Shot Error Correction of Three-Dimensional Homological Product Codes”, PRX Quantum 2, (2021) arXiv:2009.11790 DOI
- [17]
- O. Higgott and N. P. Breuckmann, “Improved Single-Shot Decoding of Higher-Dimensional Hypergraph-Product Codes”, PRX Quantum 4, (2023) arXiv:2206.03122 DOI
- [18]
- L. Berent et al., “Analog information decoding of bosonic quantum LDPC codes”, (2023) arXiv:2311.01328
- [19]
- A. Leverrier, S. Apers, and C. Vuillot, “Quantum XYZ Product Codes”, Quantum 6, 766 (2022) arXiv:2011.09746 DOI
- [20]
- 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
- [21]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [22]
- 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
- [23]
- 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

- 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