Numerical Algebraic Geometry for Energy Computations on Tensor Train Varieties

TL;DR

This paper applies computational algebraic geometry to quantum chemistry energy minimization problems on tensor train varieties, analyzing critical points, and benchmarking solution methods.

Contribution

It introduces a geometric framework for energy minimization on tensor trains, studies critical points, and benchmarks numerical algorithms using homotopy continuation.

Findings

Identified when tensor train varieties are Segre products of projective spaces.

Developed a birational parametrization from Grassmannians.

Computed critical points using homotopy continuation.

Abstract

We study energy minimization problems in quantum chemistry through the lens of computational algebraic geometry. We focus on minimizing the Rayleigh quotient of a Hamiltonian over a tensor train variety. The complex critical points of this problem approximate eigenstates of the quantum system, with the global minimum approximating the ground state. We call the number of critical points the Rayleigh-Ritz degree. After introducing tensor train varieties, we identify instances when they are Segre products of projective spaces. We also report what we know about the defining ideals of tensor trains. We present a birational parametrization of them from products of Grassmannians. 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. We use homotopy continuation to compute all critical points of this optimization problem over various tensor train and determinantal varieties. Finally, we use these results to benchmark state-of-the-art methods, the Alternating Linear Scheme and Density Matrix Renormalization Group.