  We present a method to approximate functionals $\text{Tr} \, f(A)$ of very
high-dimensional hermitian matrices $A$ represented as Matrix Product Operators
(MPOs). Our method is based on a reformulation of a block Lanczos algorithm in
tensor network format. We state main properties of the method and show how to
adapt the basic Lanczos algorithm to the tensor network formalism to allow for
high-dimensional computations. Additionally, we give an analysis of the
complexity of our method and provide numerical evidence that it yields good
approximations of the entropy of density matrices represented by MPOs while
being robust against truncations.
