Numerical Algebraic Geometry for Energy Computations

We consider an optimization problem arising in quantum chemistry. Given a real symmetric matrix H, called a Hamiltonian, and a projective variety V, we aim to minimize the Rayleigh-Ritz quotient of H over V. This corresponds to computing the ground state energy of a quantum system. The number of complex critical points of this optimization problem for a generic symmetric matrix H is known as the Rayleigh-Ritz degree of the variety V.

Using a one-to-one parameterization, we develop numerical homotopy continuation methods to compute all RR-degree many critical points for algebraic varieties of tensors with bounded tensor train rank. We compare our approach to the well-known alternating least squares (ALS) method.

This talk is based on joint work in progress with Hannah Friedman, Serkan Hoşten, and Max Pfeffer.