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

TL;DR

This paper introduces a novel four-dimensional toric code lattice based on octaplex tessellation, enabling the transversal implementation of a non-Clifford $ extsf{CCCZ}$ gate, expanding the capabilities of topological quantum codes.

Contribution

It presents a new 4D lattice structure for toric codes that allows for transversal non-Clifford gates, extending the understanding of topological codes in higher dimensions.

Findings

Octaplex tessellation supports transversal $ extsf{CCCZ}$ gate.

Generalizes the connection between dimension and transversal gates.

Differs from conventional hypercubic lattice by enabling non-Clifford gates.

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.