Exact formulas, or better approximations in minimum times in quantum control, a.k.a., the quantum speed limit is largely unsolved.

Expressed mathematically:

Given $a,b \in \mathfrak{su}(n)$ which are bracket generating and some $G \in SU(n)$, consider a system obeying the equation $\dot{U}_t = (a + f(t)b)U_t$. What is the minimum time $T=T^*$, over all controls $f$ (where all functions, or even delta functions are permitted), to achieve $U_T = G$.

Exact formulas, or better approximations in minimum times in quantum control, a.k.a., the quantum speed limit, is largely unsolved.

Expressed mathematically:

Given $a,b \in \mathfrak{su}(n)$ which are bracket generating and some $G \in SU(n)$, consider a system obeying the equation $\dot{U}_t = (a + f(t)b)U_t$. What is the minimum time $T=T^*$, over all controls $f$ (where all functions, or even delta functions are permitted), to achieve $U_T = G$.