3D fermionic surface code[14] 

Also known as 3D toric code with emergent fermion, Levin-Wen fermion model.

Description

A non-CSS 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 [5; Eq. (D45-46)]. Realizations on other lattices also exist [6,7].

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 [8,9].

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 [10].

Parents

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 [9]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [8,9].
  • 3D bosonization code — The 3D fermionic surface code is the result of applying the 3D bosonization mapping to a trivial fermonic theory [9]. Twist defects in the 3D fermionic surface code take the form of Kitaev chains after the mapping [8,9].

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]
A. Dua et al., “Sorting topological stabilizer models in three dimensions”, Physical Review B 100, (2019) arXiv:1908.08049 DOI
[6]
S. Mandal and N. Surendran, “Exactly solvable Kitaev model in three dimensions”, Physical Review B 79, (2009) arXiv:0801.0229 DOI
[7]
S. Ryu, “Three-dimensional topological phase on the diamond lattice”, Physical Review B 79, (2009) arXiv:0811.2036 DOI
[8]
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
[9]
M. Barkeshli et al., “Codimension-2 defects and higher symmetries in (3+1)D topological phases”, SciPost Physics 14, (2023) arXiv:2208.07367 DOI
[10]
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

Your contribution is welcome!

on github.com (edit & pull request)— see instructions

edit on this site

Zoo Code ID: 3d_fermionic_surface

Cite as:
“3D fermionic surface code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2024. https://errorcorrectionzoo.org/c/3d_fermionic_surface
BibTeX:
@incollection{eczoo_3d_fermionic_surface, title={3D fermionic surface code}, booktitle={The Error Correction Zoo}, year={2024}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/3d_fermionic_surface} }
Share via:
Twitter | Mastodon |  | E-mail
Permanent link:
https://errorcorrectionzoo.org/c/3d_fermionic_surface

Cite as:

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qubits/stabilizer/topological/surface/higher_d/3d_fermionic_surface.yml.