Quantum Universality in Composite Systems: A Trichotomy of Clifford Resources

The Clifford group is efficiently classically simulable, and universality is obtained by supplementing it with non-Clifford resources. We determine which single-qudit gates suffice to achieve universality. We show that the structure of such resources is governed by the prime factorization of the qudit dimension d. Using the adjoint action on the space of complex trace-zero matrices, we relate density to irreducibility together with an infiniteness criterion, yielding a trichotomy based on the factorization of d. When d is prime, any non-Clifford gate generates a dense subgroup of the determinant-one unitaries. If d is a prime power, the adjoint action is reducible, and universality requires gates that couple the resulting invariant subspaces. For composite d with pairwise coprime factors, generalized intra-qudit controlled-NOT gates connecting the factors already suffice. These findings suggest that "composite architectures" - hybrid registers combining incommensurate dimensions - offer a route to bypass the standard overhead associated with magic-state injection.