Eigenvector Continuation with Subspace Learning.

Area of Science:

• Quantum physics
• Computational physics
• Linear algebra

Background:

• Finding extremal eigenvalues and eigenvectors of large Hamiltonian matrices is crucial in quantum physics.
• Existing methods often fail when control parameters in the Hamiltonian exceed certain thresholds.

Purpose of the Study:

• To introduce a novel technique, eigenvector continuation, to extend the applicability of existing eigenvalue solvers.
• To address the challenge of computing extremal eigenvalues and eigenvectors in high-dimensional vector spaces where standard linear algebra operations are infeasible.

Main Methods:

• Eigenvector continuation leverages the low-dimensional manifold approximation of eigenvector trajectories under smooth Hamiltonian changes.
• The method is grounded in analytic function theory for theoretical validation.
• An algorithm is proposed based on these principles to solve for extremal eigenvectors.

Main Results:

• Demonstrated that eigenvector trajectories can be accurately approximated by low-dimensional manifolds.
• Successfully extended the reach of standard eigenvalue methods beyond their typical operational limits.
• Validated the technique through practical applications in quantum many-body theory.

Conclusions:

• Eigenvector continuation offers a powerful new approach for tackling challenging eigenvalue problems in quantum physics.
• The method significantly enhances computational capabilities for large-scale quantum systems.
• This technique promises broader applications in areas requiring precise eigenvector calculations.