Description
A 3D lattice stabilizer code whose low-energy excitations on boundaries realize the three-fermion anyon theory [3–5] and that can be used as a resource state for fault-tolerant MBQC [2].
Encoding
3F QCA encoder [6,7], which can be simplified using bosonization [8] and can be extended to SPTs in higher dimensions based on an exact bosonization duality [9].
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.
Parents
- Qubit stabilizer code
- 3D lattice stabilizer code
- Walker-Wang model code — The Walker-Wang model code reduces to the 3F model code when the input category \(\mathcal{C}=3F\) [2].
- Abelian topological code — When treated as ground states of the code Hamiltonian, 3F model code states realize 3F topological order, which is chiral and modular.
Cousins
- 3D bosonization code — The 3F QCA encoder [6,7] can be simplified using bosonization [8].
- Bosonization code — The 3F QCA encoder [6,7] can be extended to SPTs in higher dimensions based on an exact bosonization duality [9].
- Symmetry-protected topological (SPT) code — The 3F QCA encoder [6,7] can be extended to SPTs in higher dimensions based on an exact bosonization duality [9].
- 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].
References
- [1]
- F. J. Burnell et al., “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]
- E. Rowell, R. Stong, and Z. Wang, “On classification of modular tensor categories”, (2009) arXiv:0712.1377
- [4]
- H. Bombin, M. Kargarian, and M. A. Martin-Delgado, “Interacting anyonic fermions in a two-body color code model”, Physical Review B 80, (2009) arXiv:0811.0911 DOI
- [5]
- H. Bombin, G. Duclos-Cianci, and D. Poulin, “Universal topological phase of two-dimensional stabilizer codes”, New Journal of Physics 14, 073048 (2012) arXiv:1103.4606 DOI
- [6]
- 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
- [7]
- J. Haah, “Topological phases of unitary dynamics: Classification in Clifford category”, (2024) arXiv:2205.09141
- [8]
- L. Fidkowski and M. B. Hastings, “Pumping Chirality in Three Dimensions”, (2023) arXiv:2309.15903
- [9]
- 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