Four-dimensional toric code with non-Clifford transversal gates

The design of a four-dimensional toric code is explored with the goal finding lattice capable implementing logical $\mathsf{CCCZ}$ gate transversally. established octaplex tessellation, which regular tessellation Euclidean space whose underlying 4-cell octaplex, or hyper-diamond. This differs from conventional 4D lattice, based on hypercubic symmetric respect to $X$ and $Z$ only allows for implementation transversal Clifford gate. work further develops connection between topological dimension gates in hierarchy, generalizing known designs $\mathsf{CZ}$ $\mathsf{CCZ}$ two three dimensions, respectively.