Lowrank Riemannian eigensolver for highdimensional Hamiltonians
Abstract
Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as highdimensional eigenvalue problems, i.e., when the eigenvector can be naturally represented as a multidimensional tensor. Tensor methods have proven to be an efficient tool for the approximation of solutions of highdimensional eigenvalue problems, however, their performance deteriorates quickly when the number of eigenstates to be computed increases. We address this issue by designing a new algorithm motivated by the ideas of Riemannian optimization (optimization on smooth manifolds) for the approximation of multiple eigenstates in the tensortrain format, which is also known as matrix product state representation. The proposed algorithm is implemented in TensorFlow, which allows for both CPU and GPU parallelization.
 Publication:

Journal of Computational Physics
 Pub Date:
 November 2019
 DOI:
 10.1016/j.jcp.2019.07.003
 arXiv:
 arXiv:1811.11049
 Bibcode:
 2019JCoPh.396..718R
 Keywords:

 Matrix product state;
 TensorTrain decomposition;
 Riemannian optimization;
 Eigensolver;
 Vibrational spectra;
 Spin chains;
 Mathematics  Numerical Analysis;
 65Z05;
 15A69;
 65F15
 EPrint:
 doi:10.1016/j.jcp.2019.07.003