Iterative Power Algorithm for Global Optimization with Quantics Tensor Trains

Optimization algorithms play a central role in chemistry since optimization is the computational keystone of most molecular and electronic structure calculations. Herein, we introduce iterative power algorithm (IPA) for global formal proof convergence both discrete continuous search problems, which essential applications such as geometry optimization. IPA implements iteration method quantics tensor train (QTT) representations. Analogous to imaginary time propagation with infinite mass, starts an initial probability distribution iteratively applies recurrence relation $\rho_{k+1}({\bf x})=U({\bf x})\rho_k({\bf x})/\|U\rho_k\|_{L^1}$, where $U({\bf x})=e^{-V({\bf x})}$ defined terms potential energy surface (PES) $V({\bf x})$. Upon convergence, becomes delta function, so minimum can be obtained position expectation value. QTT representations are generated by fast adaptive interpolation multidimensional arrays bypass curse dimensionality need evaluate x})$ all possible ${\bf x}$. We illustrate capabilities two PESs, including differentiable model PES DNA chain non-differentiable PES, $V(p)=\text{mod}(N,p)$, that resolves prime factors integer $N$, $p$ space numbers folded $d$-dimensional $2_1 \times 2_2 \cdots 2_d$ tensor. find multiple degenerate minima even when separated large barriers highly rugged landscape potentials. Therefore, should great interest wide range other problems ubiquitous