Geodesic
On the approximation of functionals of very large Hermitian matrices represented as matrix product operators
Electronic transactions on numerical analysis, Tome 46 (2017), pp. 215-232.

Voir la notice de l'article provenant de la source Electronic Library of Mathematics

Summary: 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.
Classification : 65F60, 65D15, 65D30, 65F15, 46N50, 15A69
Keywords: tensor decompositions, numerical analysis, Lanczos method, Gauss quadrature, quantum physics 
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     author = {August, Moritz and Ba\~nuls, Mari Carmen and Huckle, Thomas},
     title = {On the approximation of functionals of very large {Hermitian} matrices represented as matrix product operators},
     journal = {Electronic transactions on numerical analysis},
     pages = {215--232},
     publisher = {mathdoc},
     volume = {46},
     year = {2017},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ETNA_2017__46__a3/}
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August, Moritz; Bañuls, Mari Carmen; Huckle, Thomas. On the approximation of functionals of very large Hermitian matrices represented as matrix product operators. Electronic transactions on numerical analysis, Tome 46 (2017), pp. 215-232. http://geodesic.mathdoc.fr/item/ETNA_2017__46__a3/