Thomas Huckle

Moritz August, Mari Carmen Bañuls, Thomas Huckle

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.

Moritz August, Thomas Huckle

We recently introduced a method to approximate functions of Hermitian Matrix Product Operators or Tensor Trains that are of the form $\mathsf{Tr} f(A)$. 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.