Eigenvector continuation with subspace learning

TL;DR

Eigenvector continuation is a novel method that leverages low-dimensional manifolds to efficiently find extremal eigenvalues and eigenvectors in large Hamiltonian matrices, especially when traditional methods fail beyond certain parameter thresholds.

Contribution

The paper introduces eigenvector continuation, a new technique that extends the applicability of eigenvalue algorithms by exploiting the low-dimensional structure of eigenvector trajectories.

Findings

Successfully benchmarks the method on quantum many-body examples.

Extends the range of parameters where eigenvalue problems can be solved.

Proves the low-dimensional manifold approximation using analytic function theory.

Abstract

A common challenge faced in quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix in a vector space so large that linear algebra operations on general vectors are not possible. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix exceeds some threshold value. In this work we present a new technique called eigenvector continuation that can extend the reach of these methods. The key insight is that while an eigenvector resides in a linear space with enormous dimensions, the eigenvector trajectory generated by smooth changes of the Hamiltonian matrix is well approximated by a very low-dimensional manifold. We prove this statement using analytic function theory and propose an algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.