# 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

- Category-based quantum code
- Topological code — Walker-Wang model codes can be realized using Walker-Wang model Hamiltonians, which realize 3D topological phases based on unitary braided fusion categories.

## Children

- Raussendorf-Bravyi-Harrington (RBH) cluster-state code — The Walker-Wang model code reduces to the RBH cluster-state code when the input category \(\mathcal{C}\) is that of the surface code [2; Sec. V.A].
- 3D fermionic surface code — The 3D fermionic surface code is a Walker-Wang model code with premodular input category sVec consisting of a trivial anyon and a fermion.
- Three-fermion (3F) Walker-Wang model code — The Walker-Wang model code reduces to the 3F model code when the input category \(\mathcal{C}=3F\) [2].

## 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

## Page edit log

- Victor V. Albert (2023-03-28) — most recent

## Cite as:

“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.