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

TL;DR

This paper introduces a quantum primitive called fast inversion for preconditioning linear systems, enabling efficient computation of Green's functions and matrix functions in quantum physics and chemistry.

Contribution

It presents a novel quantum preconditioning technique called fast inversion, facilitating faster solutions to quantum linear systems and matrix function evaluations.

Findings

Fast inversion effectively preconditions quantum linear systems.

Application to Green's functions in quantum many-body physics.

Development of algorithms for matrix functions like Gibbs states.

Abstract

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.