Student/Postdoc Working Geometry Seminar

Date: January 13, 2021

Time: 10:00AM - 11:00AM

Location: zoom

Speaker: V. Batista, Yale U.

Title: Tensor train methods for simulations of quantum dynamics and global optimization

Abstract: We introduce the “tensor-train split-operator Fourier transform” (TT-SOFT) algorithm for simulations of multidimensional quantum dynamics [J. Chem. Theory Comput. 13: 4034-4042 (2017)]. In the same spirit of all matrix product states, the tensor-train format enables the representation, propagation, and computation of observables of multidimensional wave functions in terms of the wavepacket tensor components in arbitrary basis sets, bypassing the need of actually computing the wave function in its full-rank tensor product space. We demonstrate the accuracy and efficiency of the TT-SOFT method as applied to propagation of 24-dimensional wave packets, describing the interconversion dynamics of pyrazine after photoexcitation into an electronically excited state. Furthermore, we introduce the iterative power algorithm (IPA) [J. Chem. Theory Comput. submitted (2021)] for global optimization, including a formal proof of convergence for both discrete and continuous optimization problems. The IPA is essentially the imaginary time propagation method with infinite mass. It is based on the power recurrence relation ρk+1(r) = Uρk/|Uρk(r)| where U = e−V (r) is defined by the scaled potential energy surface V(r), and ρk(r) is the density distribution after the k-th optimization step. We show how to implement the IPA for high-dimensional potential energy surfaces by approximating ρ(r) and V (r) in terms of low-rank quantics tensor trains (QTT) generated by fast adaptive interpolation of multidimensional arrays. The resulting QTT implementation by-passes the curse of dimensionality and the need to evaluate V(r) at all possible values of r. We illustrate the capabilities of IPA as applied to the highly rugged potential energy surface V (r) = mod(N, r) in the space of primes r {2, 3, 5, 7, 11, . . . } folded as a d-dimensional 2_1 × 2_2 × · · · × 2_d tensor. We find that IPA resolves the degenerate global minima corresponding to the prime factors of numbers N with as many as 2,773 digits, enabling the solution of