Let $U$ be a unitary matrix, and let $H$ be an Hermitian matrix. I want to know if there is a $t \in\mathbb R$ such that $\exp(i t H) = U$.

A connected question is: given a set $\{g_1, g_2, ..., g_N\}$ of Hermitian matrices, does there exist a set of real parameters $\{\lambda_1,...,\lambda_N\}$ such that $\exp\left(i\sum_k \lambda_k g_k\right) = U$?

Clearly this won't always be possible, but is there some known criterion to say something about when it would or would not be possible for a given choice of matrices?

Has this question been studied? Does this kind of problem have a name? It looks very connected to the theory of Lie algebras/groups, but because I'm looking at a specific matrix $U$ and generator $H$, as opposite to whole algebras, it also seems to be a little bit different.