  We aim to establish a scalable scheme for characterising diagonal
non-Clifford gates for single- and multi-qudit systems; \(d\) is a prime-power
integer. By employing cyclic operators and a qudit T gate, we generalise the
dihedral benchmarking scheme for single- and multi-qudit circuits. Our results
establish a path for experimentally benchmarking qudit systems and are of
theoretical and experimental interest because our scheme is optimal insofar as
it does not require preparation of the full qudit Clifford gate set to
characterise a non-Clifford gate. Moreover, combined with Clifford randomised
benchmarking, our scheme is useful to characterise the generators of a
universal gate set.
