(Sorry for poor question I don't know how to write a math equation)
$U_i$ for $i=1,2,...,d$ and $M, S$ are $C^{(N \times N)}$ matrices. And $\alpha_i$ for $i=1,2,...,d$ and $t$ are complex scalars. $H=\sum_{i=1}^d\alpha_iU_i$ and $Q=\exp(jtH)$ where $j$ is the imaginary unit - $j^2=-1$.
Let $f(\alpha_1,\alpha_2,...,\alpha_d)=\operatorname{tr}[MQSQ^H]$
Where $^H$ is the Hermitian transpose.
Then how can we describe the partial derivative of $f$ with respect to $\alpha_i$?