$$C = A^H D A + M$$
$A$ is a $m \times m$ unitary matrix.
$D$ is a $m \times m$ diagonal matrix with entries either $0$ or $1$. The number of $1$'s is $n \ll m$.
$M$ is a $m \times m$ diagonal matrix with all non-negative entries.
It is known that $C$ is a positive definite matrix. Is there a fast way to compute the lowest eigenvalue (need not compute the eigenvector) of $C$?
Especially given $n \ll m$ and $m$ being very large I cannot afford to compute all $m$ eigenvalues. Also I would like to avoid storing a $m \times m$ matrix in memory if possible.