Low-rank Riemannian eigensolver for high-dimensional Hamiltonians
This work provides a novel method for computing multiple eigenstates in quantum chemistry and spin systems, where existing tensor methods degrade in performance.
The paper addresses the problem of computing multiple eigenstates of high-dimensional Hamiltonians, where tensor methods struggle as the number of eigenstates increases. They propose a Riemannian optimization algorithm in the tensor-train format, achieving efficient parallelization on CPU and GPU.
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.