论文标题
快速反转，预处理量子线性系统求解器和矩阵函数的快速评估
Fast inversion, preconditioned quantum linear system solvers, and fast evaluation of matrix functions
论文作者
论文摘要
预处理是在经典迭代线性系统求解器的背景下处理不良条件线性系统的最广泛使用和有效的方法。我们引入了一种称为快速反转的量子原始性，该量子可用作求解量子线性系统的预处理。快速反转的关键思想是通过通过经典算术实现特征值反转的量子电路直接构造矩阵反向。我们证明了预处理线性系统求解器在计算量子多体系统的单粒子函数中的应用，这些系统广泛用于量子物理，化学和材料科学。我们在三种情况下分析了复杂性：哈伯德模型，量子二体二型二 - 基础上的量子多体性哈密顿量和schwinger模型。我们还提供了一种在固定粒子歧管中以第二量量化进行绿色功能计算的方法，并注意此方法对于更广泛的模拟可能是有价值的。除了求解线性系统外，快速反转还使我们能够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制备。我们基于轮廓积分公式和逆变换介绍了这项任务的两种有效方法。
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.
