A four-dimensional toric code with non-Clifford transversal gates

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