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