Four-dimensional toric code with non-Clifford transversal gates

@article{JochymOConnor2020FourdimensionalTC,
  title={Four-dimensional toric code with non-Clifford transversal gates},
  author={Tomas Jochym-O'Connor and Theodore J. Yoder},
  journal={arXiv: Quantum Physics},
  year={2020}
}
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… 
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