Towards a better understanding of the matrix product function approximation algorithm in application to quantum physics

We recently introduced a method to approximate functions of Hermitian Matrix Product Operators or Tensor Trains that are of the form $\mathsf{T}\mathsf{r}f(A)$. Functions of this type occur in several applications, most notably in quantum physics. In this work we aim at extending the theoretical understanding of our method by showing several properties of our algorithm that can be used to detect and correct errors in its results. Most importantly, we show that there exists a more computationally efficient version of our algorithm for certain inputs. To illustrate the usefulness of our finding, we prove that several classes of spin Hamiltonians in quantum physics fall into this input category. We finally support our findings with numerical results obtained for an example from quantum physics.