As a qubit is a two-level quantum system whose state space is spanned by |0>,

|1>, so a qudit is a d-level quantum system whose state space is spanned by

|0>,...,|d-1>. Quantum computation has stimulated much recent interest in

algorithms factoring unitary evolutions of an n-qubit state space into

component two-particle unitary evolutions. In the absence of symmetry, Shende,

Markov and Bullock use Sard's theorem to prove that at least C 4^n two-qubit

unitary evolutions are required, while Vartiainen, Moettoenen, and Salomaa

(VMS) use the QR matrix factorization and Gray codes in an optimal order

construction involving two-particle evolutions. In this work, we note that

Sard's theorem demands C d^{2n} two-qudit unitary evolutions to construct a

generic (symmetry-less) n-qudit evolution. However, the VMS result applied to

virtual-qubits only recovers optimal order in the case that d is a power of

two. We further construct a QR decomposition for d-multi-level quantum logics,

proving a sharp asymptotic of Theta(d^{2n}) two-qudit gates and thus closing

the complexity question for all d-level systems (d finite.) Gray codes are not

required, and the optimal Theta(d^{2n}) asymptotic also applies to gate

libraries where two-qudit interactions are restricted by a choice of certain

architectures.