Three-fermion (3F) Walker-Wang model code [1,2]

Cousins

• 3D bosonization code— The 3F Walker-Wang QCA encoder [4,5] can be simplified using bosonization [6].
• Bosonization code— The 3F Walker-Wang QCA encoder [4,5] can be extended to SPTs in higher dimensions based on an exact bosonization duality [7].
• Abelian topological code— The gapped boundary of the 3F Walker-Wang model supports the 3F topological order [1,2].
• 3D surface code— The 3F Walker-Wang model code is obtained from two copies of the 3D surface code on the same cubic lattice [1,3]. The copies are twisted together by decorating the face terms with \(X\)-type operators on the edges lying over and under each face [3]. Flux of one copy then confines the point excitations of the other [3].
• Symmetry-protected self-correcting quantum code— Without symmetry enforcement, the 3F Walker-Wang model code on a \(T^2\times I\) lattice stores no information, even classical, at nonzero temperature [3]. Enforcing the one-form symmetry generated by its vertex terms within a distance \(W\) of one boundary forbids string endpoints there [3]. A boundary logical string must then be completed by bulk strings carrying linearly confined flux, giving a symmetric energy barrier \(\Delta\sim\min\{L_1,L_2,W\}\) on an \(L_1\times L_2\) boundary torus [3]. The two qubits at that boundary thus form a symmetry-protected self-correcting quantum code whenever \(W\), \(L_1\), and \(L_2\) grow without bound [3]. The enforced region can have measure zero in the thermodynamic limit, e.g., for \(W\) growing logarithmically with the lattice size [3].
• 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].
• Six-semion stabilizer code— The six-semion and three-fermion anyon theories are the \(r=2\) and \(r=1\) members of the family \(F_{2^r}\) of Abelian anyon theories with fusion group \(\mathbb{Z}_{2^r}\times\mathbb{Z}_{2^r}\) [8,9].