## 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 planar 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 with bosonic charge and loop excitations (BcBl). The welded surface code does not satisfy homogeneous topological order [18].
- XYZ product code — The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [19; Exam. A.1], but done in a different way than the Chamon code; see [20; 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 [20; 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 [21], but it is not a quantum self-correcting memory [22].
- \([[8,3,2]]\) CSS code — The \([[8,3,2]]\) code can be concatenated with a 3D surface code to yield a \([[O(d^3),3,2d]]\) code family that admits a transversal implementation of the logical \(CCZ\) gate [23].
- 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 [8,24,25]. This process can be viewed as an ungauging [26–28,28] of certain symmetries. 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).
- Fractal surface code — Fractal surface codes are obtained by removing qubits from the 3D surface code on a cubic lattice.
- 3D subsystem surface code

## References

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- [2]
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- [3]
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- [4]
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- [5]
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- [18]
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- [19]
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- [20]
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- [21]
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- [22]
- Y. Li et al., “Phase diagram of the three-dimensional subsystem toric code”, (2023) arXiv:2305.06389
- [23]
- D. Hangleiter et al., “Fault-tolerant compiling of classically hard IQP circuits on hypercubes”, (2024) arXiv:2404.19005
- [24]
- 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
- [25]
- 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
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## 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