We propose a new algorithm for calculation of vibrational spectra of molecules using tensor train decomposition. Under the assumption that eigenfunctions lie on a low-parametric manifold of low-rank tensors we suggest using well-known iterative methods that utilize matrix inversion (locally optimal block preconditioned conjugate gradient method, inverse iteration) and solve corresponding linear systems inexactly along this manifold. As an application, we accurately compute vibrational spectra (84 states) of acetonitrile molecule CH3CN on a laptop in one hour using only 100 MB of memory to represent all computed eigenfunctions.
@cite_24: The density matrix renormalization group [1] (DMRG), devised to improve on the Wilson’s numerical renormalization group (NRG) [2], has become the method of choice to simulate one-dimensional quantum many-body systems at zero temperature and has found a large number of applications in the fields of the condensed matter physics, quantum chemistry and quantum information theory where it turned out that DMRG is essentially equivalent to simulating quantum systems in terms of matrix product states (MPS) [3]. In all these methods, a quantum many-body state is represented by associating matrices to local configurations at sites, and the coefficients in the expansion over the configuration states are given as products of the corresponding matrices. The DMRG can be used to calcu
@cite_2: We consider approximate computation of several minimal eigenpairs of large Hermitian matrices which come from high-dimensional problems. We use the tensor train (TT) format for vectors and matrices to overcome the curse of dimensionality and make storage and computational cost feasible. We approximate several low-lying eigenvectors simultaneously in the block version of the TT format. The computation is done by the alternating minimization of the block Rayleigh quotient sequentially for all TT cores. The proposed method combines the advances of the density matrix renormalization group (DMRG) and the variational numerical renormalization group (vNRG) methods. We compare the performance of the proposed method with several versions of the DMRG codes, and show that it may be preferable for systems with large dimension and or mode size, or when a large number of eigenstates is sought.
Alternatively to iterative methods, one can pose an optimization problem -- minimization of the Rayleigh quotient with the constraint on rank. This approach was recently considered in the matrix product state (MPS) community @cite_24 and independently in @cite_2 . The only disadvantages is that all eigenfunctions are treated in one basis, which leads to large ranks and the method becomes slow (calculation of the acetonitrile took several days). Nevertheless, this approach becomes very efficient and accurate when small number of eigenpairs are required.