## 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

The planar 3D surface code family on a cubic lattice of length \(L\) has parameters \([[2L(L-1)^2+L^3,1,d_X=L^2,d_Z=L]]\), while the 3D toric code has parameters \([[3L^3,3,d_X=L^2,d_Z=L]]\).

Stability against Hamiltonian perturbations was determined using a tensor-network representation [8]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.

## 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 [19].
- XYZ product code — The 3D planar (3D toric) code can be obtained from a hypergraph product of three repetition (cyclic) codes [20; Exam. A.1], but done in a different way than the Chamon code; see [21; 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 [21; Sec. 3.4].
- Rotated surface code — There exists a rotated version of the 3D surface code, akin to the (2D) rotated surface code [3].
- Hamiltonian-based code — Stability of the 3D surface code against Hamiltonian perturbations was determined using a tensor-network representation [8]. The phase diagram of the perturbed tensor network maps to that of a 3D Ising gauge theory.
- 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 [22], but it is not a quantum self-correcting memory [23].
- \([[8,3,2]]\) 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 [24].
- 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 [9,25,26]. This process can be viewed as an ungauging [27–29,29] of certain symmetries. This mapping can also be done via code concatenation [10].
- 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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## 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