Eigenvector continuation with subspace learning
For quantum physicists, this method solves the bottleneck of existing eigenvalue solvers failing at large control parameter values.
Eigenvector continuation extends the reach of existing methods for finding extremal eigenvalues of large Hamiltonian matrices by exploiting the low-dimensional manifold of eigenvector trajectories under smooth parameter changes, demonstrated on quantum many-body examples.
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.