Eigenvector continuation with subspace learning.

A common challenge faced in many branches of quantum physics is finding the extremal eigenvalues and eigenvectors of a Hamiltonian matrix too large to store in computer memory. There are numerous efficient methods developed for this task, but they generally fail when some control parameter in the Hamiltonian matrix such as interaction coupling exceeds some threshold value. In this work we present a new technique called eigenvector continuation that can extend the reach of these methods. Borrowing some concepts from machine learning, the key insight is that while an eigenvector resides in a linear space that may have enormous dimensions, the effective dimensionality of the eigenvector trajectory traced out by the one-parameter family of Hamiltonian matrices is small. We prove this statement using analytic function theory and propose a algorithm to solve for the extremal eigenvectors. We benchmark the method using several examples from quantum many-body theory.