Abstract
We show that single-qudit universality in Clifford-based gate sets follows a trichotomy determined by the prime factorization of the local dimension d. For prime d, any gate outside the Clifford group is universal. For prime-power dimensions d=p^m with m≥ 2, not every non-Clifford gate is universal, but it can be achieved by suitable members of a family of diagonal phase gates, generalizing the qubit T gate, as well as by permutations as simple as swapping |0 and |1 while leaving all other basis states unchanged. When d decomposes into pairwise coprime prime powers, generalized CNOT-type gates between the corresponding factors already suffice for universality. In this composite case, universality can be obtained without introducing an explicit diagonal magic gate. Our results split non-Clifford resources for high-dimensional systems into two broad mechanisms: CNOT-type (permutations) or T-type (diagonal phases) gates.