Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties: Supplementary material

Abstract

This page contains supplementary files for the paper Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties. In this article, we study energy minimization problems from quantum chemistry through the viewpoint of computational algebraic geometry. Specifically, we minimize the Rayleigh quotient of a Hamiltonian restricted to a tensor train (TT) variety. The complex critical points of this optimization problem approximate the eigenstates of a quantum system, while the global minimizer approximates the ground state. We call the number of these complex critical points the Rayleigh–Ritz (RR) degree. The paper develops several new geometric and computational tools for understanding TT varieties. We characterize situations in which such varieties are Segre products, describe their defining ideals, and provide a birational parametrization via products of Grassmannians. Using methods from numerical algebraic geometry, we compute all critical points of the Rayleigh–Ritz optimization problem for various TT and determinantal varieties. We use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group. Our experiments reveal, in small cases, the presence of multiple local minima and demonstrate that ALS may converge to suboptimal solutions. Along the way, we study the Rayleigh-Ritz degree, and we introduce the Rayleigh-Ritz discriminant, which describes Hamiltonians that lead to deficient number of critical points. The supplementary files included here contain additional data, scripts, and examples that complement the theoretical results presented in the paper.