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].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].
Primary Hierarchy
Parents
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.
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].
Chiral semion Walker-Wang model code
References
- [1]
- W. Shirley, Y.-A. Chen, A. Dua, T. D. Ellison, N. Tantivasadakarn, and D. J. Williamson, “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
Page edit log
- Victor V. Albert (2026-06-08) — most recent
- Nathanan Tantivasadakarn (2023-03-28)
- Victor V. Albert (2023-03-28)
Cite as:
“Chiral semion Walker-Wang model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2026. https://errorcorrectionzoo.org/c/3d_semion, arXiv:2606.11484