On the Approximation of Functionals of Very Large Hermitian Matrices represented as Matrix Product Operators

TL;DR

This paper introduces a tensor network-based block Lanczos method to efficiently approximate functionals of large Hermitian matrices represented as MPOs, enabling high-dimensional computations with good accuracy and robustness.

Contribution

It adapts the Lanczos algorithm to tensor networks for high-dimensional matrix functionals, providing a new computational approach with complexity analysis and numerical validation.

Findings

Effective approximation of $ ext{Tr} \, f(A)$ for large MPOs

Method is robust against truncation errors

Numerical results show good entropy approximation

Abstract

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.