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

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