Description
A 3D lattice stabilizer code whose low-energy excitations on boundaries realize a 3D time-reversal SPT order [1] and that can be used as a resource state for fault-tolerant MBQC [2]. The anyons at the boundary of the lattice are described by the 3F anyon theory.Encoding
3F QCA encoder [3,4], which can be simplified using bosonization [5] and can be extended to SPTs in higher dimensions based on an exact bosonization duality [6].Gates
Clifford gates can be performed by braiding and fusing symmetry defects in the MBQC model.Fault Tolerance
Fault-tolerant MBQC protocol by encoding in, braiding, and fusing symmetry defects.Cousins
- 3D bosonization code— The 3F Walker-Wang QCA encoder [3,4] can be simplified using bosonization [5].
- Bosonization code— The 3F Walker-Wang QCA encoder [3,4] can be extended to SPTs in higher dimensions based on an exact bosonization duality [6].
- Abelian topological code— When treated as ground states of the code Hamiltonian, 3F Walker-Wang model code states realize a 3D time-reversal SPT order [1]. The anyons at the boundary of the lattice are described by the 3F anyon theory.
- Three-fermion (3F) subsystem code— The (three-dimensional) 3F Walker-Wang model cluster-like state encodes the temporal gate operations on the (two-dimensional) 3F subsystem code into a third spatial dimension [2].
Primary Hierarchy
Parents
The Walker-Wang model code reduces to the 3F model code when the input category \(\mathcal{C}=3F\) [2]. When treated as ground states of the code Hamiltonian, 3F Walker-Wang model code states realize a 3D time-reversal SPT order [1]. The anyons at the boundary of the lattice are described by the 3F anyon theory.
Three-fermion (3F) Walker-Wang model code
References
- [1]
- F. J. Burnell, X. Chen, L. Fidkowski, and A. Vishwanath, “Exactly soluble model of a three-dimensional symmetry-protected topological phase of bosons with surface topological order”, Physical Review B 90, (2014) arXiv:1302.7072 DOI
- [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”, Communications in Mathematical Physics 406, (2025) arXiv:2205.09141 DOI
- [5]
- L. Fidkowski and M. B. Hastings, “Pumping Chirality in Three Dimensions”, (2024) arXiv:2309.15903
- [6]
- L. Fidkowski, J. Haah, and M. B. Hastings, “A QCA for every SPT”, (2024) arXiv:2407.07951
Page edit log
- Victor V. Albert (2023-03-28) — most recent
Cite as:
“Three-fermion (3F) Walker-Wang model code”, The Error Correction Zoo (V. V. Albert & P. Faist, eds.), 2023. https://errorcorrectionzoo.org/c/three_fermion