## Description

Lattice stabilizer code in three spatial dimensions. Qubit codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code via a local constant-depth Clifford circuit [1].

## Code Capacity Threshold

Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, and SFSL code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [2].

## Parents

## Children

## Cousins

- Clifford-deformed surface code (CDSC) — Applying Clifford deformations to various 3D stabilizer codes, including the 3D surface code, 3D color code, X-cube model code, the SFSL code, yields a \(50\%\) code capacity threshold under infinitely biased Pauli noise [2].
- Abelian topological code — Qubit 3D stabilizer codes are conjectured to admit either fracton phases or abelian topological phases that are equivalent to multiple copies of the 3D surface code and/or the 3D fermionic surface code [1].
- Self-correcting quantum code — 3D translationally-invariant qubit stabilizer code families with constant \(k\) support logical string operators and thus cannot be self-correcting [3]. For non-constant \(k\), such families can support at most a logarithmic energy barrier [4].

## References

- [1]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [2]
- E. Huang et al., “Tailoring Three-Dimensional Topological Codes for Biased Noise”, PRX Quantum 4, (2023) arXiv:2211.02116 DOI
- [3]
- B. Yoshida, “Feasibility of self-correcting quantum memory and thermal stability of topological order”, Annals of Physics 326, 2566 (2011) arXiv:1103.1885 DOI
- [4]
- J. Haah, “Local stabilizer codes in three dimensions without string logical operators”, Physical Review A 83, (2011) arXiv:1101.1962 DOI

## Page edit log

- Victor V. Albert (2024-01-27) — most recent

## Cite as:

“3D lattice stabilizer code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/3d_stabilizer