Chiral semion Walker-Wang model code[1] 

Description

A 3D lattice modular-qudit stabilizer code with qudit dimension \(q=4\) whose low-energy excitations on boundaries realize the chiral semion topological order. The model admits 2D chiral semion topological order at one of its surfaces [1,2]. The corresponding phase can also be realized via a non-stabilizer Hamiltonian [3].

Encoding

A unitary QCA encoder applied to product state realizes the 3D chiral semion Walker-Wang model code, which in turn admits 2D chiral semion topological order if truncated at one of its surfaces [1,2].

Parents

  • Modular-qudit stabilizer code
  • 3D lattice stabilizer code
  • Walker-Wang model code — The Walker-Wang model code reduces to the chiral semion model code when the input category is \(\mathcal{C}=\mathbb{Z}_{2}^{(1/2)}\), or alternatively \(\mathcal{C}=\mathbb{Z}_{4}^{(1)}\) after condensing a \(\mathbb{Z}_{2}\)-transparent boson.
  • Dijkgraaf-Witten gauge theory code — When treated as ground states of the code Hamiltonian, the code states realize 3D double-semion topological order, a topological phase of matter that exists as the deconfined phase of the 3D twisted \(\mathbb{Z}_2\) gauge theory [4].

Cousin

  • Chiral semion subsystem code — A unitary QCA encoder applied to product state realizes the 3D chiral semion Walker-Wang model code, which in turn admits 2D chiral semion topological order if truncated at one of its surfaces [1,2].

References

[1]
W. Shirley et al., “Three-Dimensional Quantum Cellular Automata from Chiral Semion Surface Topological Order and beyond”, PRX Quantum 3, (2022) arXiv:2202.05442 DOI
[2]
J. Haah, “Clifford quantum cellular automata: Trivial group in 2D and Witt group in 3D”, Journal of Mathematical Physics 62, (2021) arXiv:1907.02075 DOI
[3]
C. W. von Keyserlingk, F. J. Burnell, and S. H. Simon, “Three-dimensional topological lattice models with surface anyons”, Physical Review B 87, (2013) arXiv:1208.5128 DOI
[4]
R. Dijkgraaf and E. Witten, “Topological gauge theories and group cohomology”, Communications in Mathematical Physics 129, 393 (1990) DOI
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Zoo Code ID: 3d_semion

Cite as:
“Chiral semion Walker-Wang model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/3d_semion
BibTeX:
@incollection{eczoo_3d_semion, title={Chiral semion Walker-Wang model code}, booktitle={The Error Correction Zoo}, year={2023}, editor={Albert, Victor V. and Faist, Philippe}, url={https://errorcorrectionzoo.org/c/3d_semion} }
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Cite as:

“Chiral semion Walker-Wang model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/3d_semion

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/qudits/stabilizer/topological/3d_semion.yml.