A quadratic Grassmann manifold optimization problem arising from quantum embedding methods

TL;DR

This paper analyzes a quadratic optimization problem on the Grassmann manifold relevant to quantum embedding, proposing numerical strategies and identifying cases where global solutions are accessible, thereby enhancing existing algorithms.

Contribution

It introduces a mathematical framework for solving a non-convex quadratic problem on the Grassmann manifold, with methods to find global minima or good initializations for quantum embedding applications.

Findings

Global minimizer can be found via an auxiliary convex problem in certain cases

Auxiliary problem solutions improve Riemannian optimization and SCF algorithms

Application to quantum embedding methods for constructing bath orbitals

Abstract

This article presents a mathematical analysis and numerical strategies for solving the optimization problem of minimizing the quadratic function $J(P) = \text{Tr}(BP)- \frac{1}{2} \text{Tr}(A P A P)$, where $A,B \in \mathbb R^{M \times M}_{\rm sym}$, with $A \succeq 0$, over the Grassmann manifold ${\rm Gr}(m,\mathbb R^M)$. While this problem is non-convex and typically admits non-global local minima - posing challenges for Riemannian optimization and self-consistent field (SCF) algorithms - we identify cases where the global minimizer can be obtained by solving an auxiliary convex problem. When this approach is not directly applicable, the solution to the auxiliary problem still serves as an effective initialization for Riemannian optimization methods and SCF algorithms, significantly improving their performance. This work is motivated by applications in quantum embedding methods, particularly in the construction of bath orbitals, where such optimization problems naturally arise.