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.
Highlights
We have established the existence of a 4D topological code with a transversal CCCZ gate
The natural candidate in 5D would be to consider lattices with Schläfli symbol: {3, 3, 4, a, b}, where a, b are integer degrees of freedom. This choice is motivated by the fact that underlying 4-cells composing the lattice are hyperoctahedra, which are the 4D analog of the octahedron
The only regular tessellation of 5D Euclidean space is from the hypercubic family: {4, 3, 3, 3, 4}, and no regular tessellation with the required conditions exist
Summary
Quantum error correction is expected to play an essential role in the development of large-scale quantum computers. Topological error correcting codes are among the most well-studied forms of quantum error correcting codes These codes are defined by their spatially local stabilizer checks in D dimensions, while embedding their logical information in macroscopic non local degrees of freedom. Such codes provide numerous computing advantages, including a pathway for experimental qubit layout [2,3,4,5,6], efficient decoding algorithms [7,8,9,10,11,12], and provable target threshold error rates for numerous local noise models [2,13,14]. Topological codes in D dimensions will be limited to having transversal gates that are in the Dth level of the Clifford hierarchy [21]. VI provides a discussion on the difficulty in finding higher-dimensional generalizations and other open questions
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