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arXiv · 1811.11049

Low-rank Riemannian eigensolver for high-dimensional Hamiltonians

Maxim Rakhuba·Alexander Novikov·Ivan Oseledets
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Abstract

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.

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Maxim Rakhuba, Alexander Novikov, Ivan Oseledets. 2018-11-27. Low-rank Riemannian eigensolver for high-dimensional Hamiltonians. https://doi.org/10.1016/j.jcp.2019.07.003

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