Randomised benchmarking for universal qudit gates

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.