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 plane-wave-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. Aside from 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.

中文翻译：

---

快速反演、预处理的量子线性系统求解器、快速格林函数计算和矩阵函数的快速评估

在经典迭代线性系统求解器的上下文中，预处理是处理病态线性系统的最广泛使用和最有效的方法。我们引入了一种称为快速反演的量子原语，它可以用作求解量子线性系统的预处理器。快速求逆的关键思想是通过经典算法实现特征值求逆的量子电路直接对矩阵求逆进行块编码。我们展示了预处理线性系统求解器在计算量子多体系统的单粒子格林函数中的应用，这些函数广泛用于量子物理、化学和材料科学。我们分析了三种情况的复杂性：哈伯德模型、平面波对偶基中的量子多体哈密顿量和施温格模型。我们还提供了一种在固定粒子流形内的二次量化中执行格林函数计算的方法，并注意到这种方法可能对更广泛的模拟有价值。除了求解线性系统，快速反演还允许我们开发用于计算矩阵函数的快速算法，例如吉布斯状态的有效准备。我们分别基于轮廓积分公式和逆变换为此类任务引入了两种有效方法。例如吉布斯状态的有效制备。我们分别基于轮廓积分公式和逆变换为此类任务引入了两种有效方法。例如吉布斯状态的有效制备。我们分别基于轮廓积分公式和逆变换为此类任务引入了两种有效方法。