Walker-Wang model code[1] 

Description

A 3D topological code defined by a unitary braided fusion category \( \mathcal{C} \) (also known as a unitary premodular category). The code is defined on a cubic lattice that is resolved to be trivalent, with a qudit of dimension \( |\mathcal{C}| \) located at each edge. The codespace is the ground-state subspace of the Walker-Wang model Hamiltonian [1]. A single-state version of the code provides a resource state for MBQC [2].

Encoding

For modular chiral anyon theories, a unitary encoder is conjectured to not be implementable in constant depth because it is believed to be an example of a quantum cellular automaton (QCA) (i.e., causal or locality-preserving automorphism) that cannot be locally implemented [3,4]. States of modular gapped theories can be initialized in constant depth [5].

Parents

Children

Cousins

  • String-net code — The Walker-Wang model is a generalization of the 3D version of the Levin-Wen model [6; Sec. 5], which realizes gauge theories coupled with bosons and fermions.
  • Abelian topological code — Any Abelian anyon theory \(A\) can be realized at one of the surfaces of a 3D Walker-Wang model whose underlying theory is an Abelian TQD containing \(A\) as a subtheory [7,8][9; Appx. H].

References

[1]
K. Walker and Z. Wang, “(3+1)-TQFTs and Topological Insulators”, (2011) arXiv:1104.2632
[2]
S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
[3]
J. Haah, L. Fidkowski, and M. B. Hastings, “Nontrivial Quantum Cellular Automata in Higher Dimensions”, Communications in Mathematical Physics 398, 469 (2022) arXiv:1812.01625 DOI
[4]
J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2024) arXiv:2205.09141
[5]
A. Bauer, “Disentangling modular Walker-Wang models via fermionic invertible boundaries”, Physical Review B 107, (2023) arXiv:2208.03397 DOI
[6]
M. A. Levin and X.-G. Wen, “String-net condensation: A physical mechanism for topological phases”, Physical Review B 71, (2005) arXiv:cond-mat/0404617 DOI
[7]
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
[8]
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
[9]
T. D. Ellison et al., “Pauli topological subsystem codes from Abelian anyon theories”, Quantum 7, 1137 (2023) arXiv:2211.03798 DOI
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Zoo Code ID: walker_wang

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

Github: https://github.com/errorcorrectionzoo/eczoo_data/edit/main/codes/quantum/categories/walker_wang.yml.