Also known as 3D toric code with emergent fermion.

## Description

A 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion. The model can be defined on a cubic lattice in several ways [4; Eq. (D45-46)]

3D fermionic 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 fermionic surface code.

## Transversal Gates

CCZ and CS gates can be obtained for the fermionic 3D surface code on certain manifolds by circuits that can be interpreted as moving and spreading lattice realizations of Kitaev chain and \(p+ip\) defects [5].

## Parents

- Generalized surface code
- 3D lattice stabilizer code
- Abelian topological code — The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with an emergent fermion.

## References

- [1]
- M. Levin and X.-G. Wen, “Fermions, strings, and gauge fields in lattice spin models”, Physical Review B 67, (2003) arXiv:cond-mat/0302460 DOI
- [2]
- K. Walker and Z. Wang, “(3+1)-TQFTs and Topological Insulators”, (2011) arXiv:1104.2632
- [3]
- F. J. Burnell et al., “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order”, Physical Review B 90, (2014) arXiv:1302.7072 DOI
- [4]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [5]
- M. Barkeshli, P.-S. Hsin, and R. Kobayashi, “Higher-group symmetry of (3+1)D fermionic \(\mathbb{Z}_2\) gauge theory: logical CCZ, CS, and T gates from higher symmetry”, (2023) arXiv:2311.05674

## Page edit log

- Victor V. Albert (2023-11-27) — most recent

## Cite as:

“3D fermionic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/3d_fermionic_surface