Efficient quantum gate teleportation in higher dimensions

The Clifford hierarchy is a nested sequence of sets quantum gates critical to achieving fault-tolerant computation. Diagonal the and 'nearly diagonal' semi-Clifford are particularly important: they admit efficient gate teleportation protocols that implement these with fewer ancillary resources such as magic states. Despite practical importance gates, many questions about their structure remain open; this especially true in higher-dimensional qudit setting. Our contribution leverage discrete Stone-von Neumann theorem symplectic formalism stabiliser mechanics towards extending results Zeng-Cheng-Chuang (2008) Beigi-Shor (2010) higher dimensions uniform manner. We further give simple algorithm for recursively enumerating all hierarchy, recognising diagonalising concise proof classification diagonal due Cui-Gottesman-Krishna (2016) single-qudit case. generalise setting prove every third level one (of any prime dimension) two qutrits can be implemented efficiently. Numerical evidence gathered via aforementioned algorithms support conjecture higher-level