  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. We analyze the complexities in three scenarios: the Hubbard model, the
quantum many-body Hamiltonian in the planewave-dual basis, and the Schwinger
model. We also provide a method for performing Green's function calculation in
second quantization within a fixed particle manifold and note that this
approach may be valuable for simulation more broadly. Besides solving linear
systems, fast inversion also allows us to develop fast algorithms for computing
matrix functions, such as the efficient preparation of Gibbs states. We
introduce two efficient approaches for such a task, based on the contour
integral formulation and the inverse transform respectively.
