Low-rank Riemannian eigensolver for high-dimensional Hamiltonians

Journal of Computational Physics. 2019. Vol. 396. P. 718–737.

Rakhuba Maxim, Novikov A., Oseledets I.

Such problems as computation of spectra of spin chains and vibrational spectra of molecules can be written as high-dimensional 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 high-dimensional 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 tensor-train 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.

Keywords: Riemannian optimization, Matrix product state, Tensor-Train decomposition, eigensolver, Vibrational spectra, Spin chains