Color code with a logical control-$S$ gate using transversal $T$ rotations

Abstract

The color code has been invaluable for the development of the theory of fault-tolerant logic gates using transversal rotations. Three-dimensional examples of the color code have shown us how its structure, specifically the intersection of the supports of logical operators, can give rise to non-Clifford $T$ and $CCZ$ gates. Here we present a color code with a logical control-$S$ gate that is accomplished with transversal $T$ and $T^{†}$ rotations on its physical qubits.

7 pages, 2 figures, comments welcome; v2 - additional supplemental information in response to peer review request