# 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] and realizes the Crane-Yetter model [2–4]. A single-state version of the code provides a resource state for MBQC [5].

## Protection

Codespace dimensions (i.e., ground-state degeneracy) has been calculated for various boundary conditions [6].

## 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 [7,8]. States of modular gapped theories can be initialized in constant depth [9].

## Parent

- \(G\)-enriched Walker-Wang model code — \(G\)-enriched Walker-Wang models reduce to Walker-Wang models for trivial \(G\) [10].

## 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 [5; Sec. V.A].
- 3D fermionic surface code — The 3D fermionic surface code is a Walker-Wang model code with premodular input category \(\mathcal{C} = \text{sVec}\) 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 \(\mathcal{C}=3F\) [5].
- Chiral semion 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.

## Cousins

- String-net code — The Walker-Wang model is a generalization of the 3D version of the Levin-Wen model [11; Sec. 5], which realizes gauge theories coupled to bosons and fermions.
- Two-gauge theory code — Two-gauge theory codes for particular two-groups are dual to certain Walker-Wang models based on Abelian groups [12; Sec. V][13; Sec. 7].
- 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 [14,15][16; Appx. H].

## References

- [1]
- K. Walker and Z. Wang, “(3+1)-TQFTs and Topological Insulators”, (2011) arXiv:1104.2632
- [2]
- L. Crane and D. N. Yetter, “A categorical construction of 4D TQFTs”, (1993) arXiv:hep-th/9301062
- [3]
- L. Crane, L. H. Kauffman, and D. N. Yetter, “Evaluating the Crane-Yetter Invariant”, (1993) arXiv:hep-th/9309063
- [4]
- L. Crane, L. H. Kauffman, and D. N. Yetter, “State-Sum Invariants of 4-Manifolds I”, (1994) arXiv:hep-th/9409167
- [5]
- S. Roberts and D. J. Williamson, “3-Fermion Topological Quantum Computation”, PRX Quantum 5, (2024) arXiv:2011.04693 DOI
- [6]
- 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
- [7]
- 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
- [8]
- J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2024) arXiv:2205.09141
- [9]
- A. Bauer, “Disentangling modular Walker-Wang models via fermionic invertible boundaries”, Physical Review B 107, (2023) arXiv:2208.03397 DOI
- [10]
- D. J. Williamson and Z. Wang, “Hamiltonian models for topological phases of matter in three spatial dimensions”, Annals of Physics 377, 311 (2017) arXiv:1606.07144 DOI
- [11]
- 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
- [12]
- A. Bullivant et al., “Topological phases from higher gauge symmetry in3+1dimensions”, Physical Review B 95, (2017) arXiv:1606.06639 DOI
- [13]
- C. Delcamp and A. Tiwari, “On 2-form gauge models of topological phases”, Journal of High Energy Physics 2019, (2019) arXiv:1901.02249 DOI
- [14]
- 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
- [15]
- 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
- [16]
- 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