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arXiv · 2010.02238

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

Tomas Jochym-O'Connor·Theodore J. Yoder
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Abstract

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.

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Tomas Jochym-O'Connor, Theodore J. Yoder. 2020-10-05. A four-dimensional toric code with non-Clifford transversal gates. https://doi.org/10.1103/physrevresearch.3.013118

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