  We generalize an efficient decomposition method for diagonal operators by
Welch et al. to qudit systems. The phase-context aware method focusses on
cascaded entanglers whose decomposition into multi-controlled INC-gates can be
optimized by the choice of a proper signed base-$d$ representation for the
natural numbers.
  While the gate count of the best known decomposition method for general
diagonal operators on qubit systems scales with $\mathcal{O}(2^n)$, the
circuits synthesized by the Welch algorithm for diagonal operators with $k$
distinct phases are upper-bounded by $\mathcal{O}(n^2k)$, which is generalized
to $\mathcal{O}(dn^2k)$ for the qudit case in this paper.
