快速反演，预处理量子线性系统解算器，以及矩阵函数的快速求值
Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions
摘要：
在经典迭代线性系统求解器中，预处理是处理病态线性系统最广泛和有效的方法。我们引入了一种称为快速反演的量子基元，它可以作为求解量子线性系统的预条件子。快速求逆的关键思想是通过一个量子线路直接对矩阵的逆矩阵进行块编码，通过经典算法实现特征值的求逆。我们演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，量子多体系统广泛应用于量子物理，化学和材料科学。我们分析了三种情况下的复杂性：哈伯德模型，量子多体哈密顿在平面波对偶基础上，和Schwinger模型。我们还提供了一种方法，用于执行格林函数计算在第二量子化内的一个固定的粒子流形，并指出，这种方法可能是有价值的模拟更广泛。除了求解线性系统之外，快速求逆还使我们能够开发计算矩阵函数的快速算法，例如有效地准备吉布斯状态。我们介绍了两种有效的方法，这样的任务，分别基于轮廓积分公式和逆变换。
英文摘要：
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.
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