## Description

A non-CSS 3D Kitaev surface code that realizes \(\mathbb{Z}_2\) gauge theory with an emergent fermion, i.e., the fermionic-charge bosonic-loop (FcBl) phase [5]. The model can be defined on a cubic lattice in several ways [6; Eq. (D45-46)]. Realizations on other lattices also exist [7,8].

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. However, unlike the 3D surface code, an open (a.k.a. rough) boundary is not possible. Twist defects in the form of Kitaev chains can be introduced as in the 2D surface code to store additional logicals [9,10].

## Transversal Gates

## Parents

- Walker-Wang model code — The 3D fermionic surface code is a Walker-Wang model code with premodular input category \(\mathcal{C} = \text{sVec}\) consisting of a trivial anyon and a fermion.
- 3D lattice stabilizer code
- Abelian topological code — The 3D Kitaev surface code realizes 3D \(\mathbb{Z}_2\) gauge theory with fermionic charge and bosonic loop excitations (FcBl), i.e., with an emergent fermion.

## Cousins

- 3D 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).
- Kitaev chain code — The 3D fermionic surface code is the result of applying the 3D bosonization mapping to a trivial fermonic theory [10]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [9,10].
- 3D bosonization code — The 3D fermionic surface code is the result of applying the 3D bosonization mapping to a trivial fermonic theory [10]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [9,10].

## 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]
- J. Haah, “Commuting Pauli Hamiltonians as Maps between Free Modules”, Communications in Mathematical Physics 324, 351 (2013) arXiv:1204.1063 DOI
- [4]
- Y.-A. Chen and A. Kapustin, “Bosonization in three spatial dimensions and a 2-form gauge theory”, Physical Review B 100, (2019) arXiv:1807.07081 DOI
- [5]
- L. Fidkowski, J. Haah, and M. B. Hastings, “Gravitational anomaly of (3+1) -dimensional Z2 toric code with fermionic charges and fermionic loop self-statistics”, Physical Review B 106, (2022) arXiv:2110.14654 DOI
- [6]
- A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
- [7]
- S. Mandal and N. Surendran, “Exactly solvable Kitaev model in three dimensions”, Physical Review B 79, (2009) arXiv:0801.0229 DOI
- [8]
- S. Ryu, “Three-dimensional topological phase on the diamond lattice”, Physical Review B 79, (2009) arXiv:0811.2036 DOI
- [9]
- P. Webster and S. D. Bartlett, “Fault-tolerant quantum gates with defects in topological stabilizer codes”, Physical Review A 102, (2020) arXiv:1906.01045 DOI
- [10]
- M. Barkeshli et al., “Codimension-2 defects and higher symmetries in (3+1)D topological phases”, SciPost Physics 14, (2023) arXiv:2208.07367 DOI
- [11]
- 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”, SciPost Physics 16, (2024) arXiv:2311.05674 DOI

## Page edit log

- Nathanan Tantivasadakarn (2024-03-26) — most recent
- Victor V. Albert (2023-11-27)

## Cite as:

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