Suppose I am given a universal set of quantum gates $S_u$ (e.g. the single-qubit rotation gate $U$ and the controlled-not gate $CX$). This is known to be universal, so any quantum circuit can be decomposed in a sequence of operations $\in S_u$. However, this process of compiling/transpiling the circuit can in general raise the depth of the original circuit.
So here is my question: is it possible to extend the set $S_u$ in order to transpile any kind of quantum circuit to its corresponding minimal-depth version? Of course, adding more and more gates to the set will possibly reduce the circuit depth but what I want is the "minimal" $S_u^*$, so that adding more gates to it will not reduce the depth anymore.
EDIT:
To be more clear about what I really need to do, consider the following example: suppose we are given the circuit $SWAP_{0,1}$ (single gate, depth=1) and the universal set of gates $S_u = \left\{U, CX\right\}$. Of course, we know that any quantum circuit can be transpiled by a proper sequence of gates $\in S_u$. However, $SWAP_{0,1}$ would be transformed into $CX_{0,1} \cdot CX_{1,0} \cdot CX_{0,1}$, which is a depth=3 circuit. So, in this trivial case, I can extend my set to $S_u = \left\{U, CX, SWAP\right\}$ in order to be sure that transpiling the circuit will lead again to depth=1.
In general, what I want is an extended (but "minimal") universal set $S_u^*$ that I can use to transpile any given quantum circuit without making it deeper (or better, making it as shallow as possible).