  Quantum circuits of a general quantum gate acting on multiple $d$-level
quantum systems play a prominent role in multi-valued quantum computation. We
first propose a new recursive Cartan decomposition of semi-simple unitary Lie
group $U(3^n)$ (arbitrary $n$-qutrit gate). Note that the decomposition
completely decomposes an n-qutrit gate into local and non-local operations. We
design an explicit quantum circuit for implementing arbitrary two-qutrit gates,
and the cost of our construction is 21 generalized controlled X (GCX) and
controlled increment (CINC) gates less than the earlier best result of 26 GGXs.
Moreover, we extend the program to the $n$-qutrit system, and the quantum
circuit of generic $n$-qutrit gates contained
$\frac{41}{96}\cdot3^{2n}-4\cdot3^{n-1}-(\frac{n^2}{2}+\frac{n}{4}-\frac{29}{32})$
GGXs and CINCs is presented. Such asymptotically optimal structure is the best
known result so far.
