Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions

Understand

Preconditioning is the most widely used and effective way for treating ill-conditioned linear systems in the context of classical iterative linear system solvers.

• We introduce a quantum primitive called fast inversion, which can be used as a preconditioner for solving quantum linear systems.
• The key idea of fast inversion is to directly block-encode a matrix inverse through a quantum circuit implementing the inversion of eigenvalues via classical arithmetics.
• We demonstrate the application of preconditioned linear system solvers for computing single-particle Green's functions of quantum many-body systems, which are widely used in quantum physics, chemistry, and materials science.