Convexity of complex matrix exponentiation

Question

I would like to evaluate the convexity of the function $f(x)$ such that $f:\mathbb{C}^{2n}\rightarrow\mathbb{R}$, where \begin{equation} f(x) = \text{Tr}\left[\text{exp}\left(i\pmatrix{A &&B\\-B^\dagger&&-C}\right)^\dagger \text{exp}\left(i\pmatrix{A &&B\\-B^\dagger&&-C}\right)\right], \end{equation} where $A$ and $C$ are real predefined $n \times n$ diagonal matrices and $B$ a complex Hankel matrix with $x$ as its content, where $x \in \mathbb{C}^{2n}$ (so that $F\in \mathbb{C}^n$). Normally, $n$ is very big (say in the order of 100). The expression $\text{exp}(iQ)$ is the matrix exponentiation such that $\text{exp}(iQ) = \sum_{n = 0}^\inf \frac{(iQ)^n}{n!}$. The convexity of the trace seems fairly easy to me, as it is just a norm. But I have no idea about how to evaluate the matrix exponentiation part. Does anyone have an idea?