## On the approximation of functionals of very large Hermitian matrices represented as matrix product operators

Moritz August, Mari Carmen Bañuls, and Thomas Huckle

### Abstract

We present a method to approximate functionals $\mathsf{Tr} f(A)$ 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.

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### Key words

tensor decompositions, numerical analysis, Lanczos method, Gauss quadrature, quantum physics

### AMS subject classifications

65F60, 65D15, 65D30, 65F15, 46N50, 15A69

### Links to the cited ETNA articles

 [5] Vol. 2 (1994), pp. 1-21 D. Calvetti, L. Reichel, and D. C. Sorensen: An implicitly restarted Lanczos method for large symmetric eigenvalue problems [8] Vol. 33 (2008-2009), pp. 207-220 L. Elbouyahyaoui, A. Messaoudi, and H. Sadok: Algebraic properties of the block GMRES and block Arnoldi methods

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