Iterative Power Algorithm for Global Optimization with Quantics Tensor Trains

Abstract

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce the iterative power algorithm (IPA) for global optimization and a formal proof of convergence for both discrete and continuous global search problems, which is essential for applications in chemistry such as molecular geometry optimization. IPA implements the power iteration method in quantics tensor train (QTT) representations. Analogous to the imaginary time propagation method with infinite mass, IPA starts with an initial probability distribution ρ0(x) and iteratively applies the recurrence relation ρ k+1(x) = U(x) ρ k (x)/∥Uρ k ∥ L 1 , where U(x) = e–V(x) is defined in terms of the potential energy surface (PES) V(x) with global minimum at x = x*. Upon convergence, the probability distribution becomes a delta function δ­(x – x*), so the global minimum can be obtained as the position expectation value x* = Tr­[x δ­(x – x*)]. QTT representations of V(x) and ρ­(x) are generated by fast adaptive interpolation of multidimensional arrays to bypass the curse of dimensionality and the need to evaluate V(x) for all possible values of x. We illustrate the capabilities of IPA for global search optimization of two multidimensional PESs, including a differentiable model PES of a DNA chain with D = 50 adenine–thymine base pairs, and a discrete non-differentiable potential energy surface, V(p) = mod­(N,p), that resolves the prime factors of an integer N, with p in the space of prime numbers {2, 3,..., p max} folded as a d-dimensional 21 × 22 × ··· × 2 d tensor. We find that IPA resolves multiple degenerate global minima even when separated by large energy barriers in the highly rugged landscape of the potentials. Therefore, IPA should be of great interest for a wide range of other optimization problems ubiquitous in molecular and electronic structure calculations.

Tensor-Train Split-Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations

“…have been defined according to the model presented in ref 54 in place of the standard harmonic fit to the electronic ground potential energy surface at the equilibrium geometry in order to facilitate direct comparison to literature results. The wavepacket then evolves 37,72,73 The TT format of an arbitrary d-dimensional tensor…”

Quantum optimal control in quantum technologies. Strategic report on current status, visions and goals for research in Europe

“…A modified version of GRAPE, based on a Krylov approximation of the matrix exponential, allows for dealing with high-dimensional Hilbert spaces [355]. A global optimization algorithm with quantics tensor trains has been proposed [524]. Improved convergence is obtained when including second order derivative information.…”

Tensor-Train Split Operator KSL (TT-SOKSL) Method for Quantum Dynamics Simulations

“…TT-SOKSL relies on the tensor-train (TT) format, 61-64 also called matrix product states (MPS) with open boundary conditions, [65][66][67][68][69] recently explored for the development of methods for quantum dynamics and global optimization. 35,70,71 The TT format of an arbitrary d-dimensional tensor X ∈ C n 1 ×...×n d involves a train-like matrix product of d 3-mode tensors X i ∈ C r i−1 ×n i ×r i with r 0 = r d = 1, so any element X(j 1 , ..., j d ) of X can be evaluated, as follows: 61…”