Functional Tensor-Train Chebyshev Method for Multidimensional Quantum Dynamics Simulations

Abstract

Methods for efficient simulations of multidimensional quantum dynamics are essential for theoretical studies of chemical systems where quantum effects are important, such as those involving rearrangements of protons or electronic configurations. Here, we introduce the functional tensor-train Chebyshev (FTTC) method for rigorous nuclear quantum dynamics simulations. FTTC is essentially the Chebyshev propagation scheme applied to the initial state, represented in continuous analogue tensor-train format. We demonstrate the capabilities of FTTC as applied to simulations of proton quantum dynamics in a 50-dimensional model of hydrogen-bonded DNA base pairs.

Fast global spectral methods for three-dimensional partial differential equations

“…For certain problems these tensors admit good low-rank approximations. Applying such approximations leads to so called functional low-rank approximations [6,15,25,30,53], which have the potential to drastically reduce the storage complexity. Exploiting this potential requires determining suitable approximation formats and using a specialized solver for the chosen formats.…”

“…For instance, in [52] it is shown that under certain conditions the solution of Laplace-like equations ( 17) can be represented in tensor train or Tucker format when the right hand side is given in the same format. The functional tensor train format [6,11,25,53], could potentially be used to extend the global spectral method to higher dimensional, linear PDEs on hypercubes. The computational complexity of time-dependent problems could be reduced using rank-adaptive, dynamical low-rank approximations [3,13,14,35].…”