# Layer code[1]

## Description

Member of a family of 3D lattice CSS codes with stabilizer generator weights \(\leq 6\) that are obtained by coupling layers of 2D surface code according to the Tanner graph of a QLDPC code. Geometric locality is maintained because, instead of being concatenated, each pair of parallel surface-code squares are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.

## Rate

Code parameters on a cube, \([[n,\Theta(n^{1/3}),\Theta(n^{1/3})]]\), achieve the 3D BPT bound when asymptotically good QLDPC codes are used in the construction.

## Parents

- Fracton stabilizer code — Layer codes are non-translation invariant 3D lattice stabilizer codes that can be viewed as fracton topological defect networks [1].
- Abelian topological code — The Layer code realizes 2D layers of \(\mathbb{Z}_2\) gauge theory coupled along defects.

## Cousins

- Good QLDPC code — Layer code parameters, \([[n,\Theta(n^{1/3}),\Theta(n^{1/3})]]\), achieve the BPT bound in 3D when asymptotically good QLDPC codes are used in the construction.
- Concatenated quantum code — Each pair of surface-code squares in a layer code are fused (or quasi-concatenated) with perpendicular surface-code squares via lattice surgery.
- Self-correcting quantum code — The energy barrier for layer-code excitations for codes constructed using asymptotically good QLDPC codes scales as \(\Theta{n^{1/3}}\).
- Kitaev surface code — Layer codes are combinations of constant-rate QLDPC codes with surface codes and build using lattice surgery.

## References

- [1]
- D. J. Williamson and N. Baspin, “Layer Codes”, (2023) arXiv:2309.16503

## Page edit log

- Victor V. Albert (2024-02-12) — most recent

## Cite as:

“Layer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/layer