Computation of extreme eigenvalues in higher dimensions using block tensor train format

TL;DR

This paper introduces a block tensor train method for efficiently computing multiple minimal eigenvalues of large high-dimensional Hermitian matrices, overcoming computational challenges in high-dimensional eigenproblems.

Contribution

The paper presents a novel block tensor train approach for simultaneous eigenpair computation, improving efficiency over traditional sequential methods in high-dimensional settings.

Findings

The block TT method effectively computes multiple eigenstates.

Compared to deflation, the block TT approach is more efficient for multiple eigenvalues.

Numerical examples demonstrate the method's advantages over quantum physics algorithms.

Abstract

We consider an approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high--dimensional problems. We use the tensor train format (TT) for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. Applying a block version of the TT format to several vectors simultaneously, we compute the low--lying eigenstates of a system by minimization of a block Rayleigh quotient performed in an alternating fashion for all dimensions. For several numerical examples, we compare the proposed method with the deflation approach when the low--lying eigenstates are computed one-by-one, and also with the variational algorithms used in quantum physics.