Solving systems of linear equations is a fundamental problem, but it can be computationally intensive for classical algorithms in high dimensions. Existing quantum algorithms can achieve exponential speedups for the quantum linear system problem (QLSP) in terms of the problem dimension, but even such a theoretical advantage is bottlenecked by the condition number of the coefficient matrix. In this work, we propose a new quantum algorithm for QLSP inspired by the classical proximal point algorithm (PPA). Our proposed method can be viewed as a meta-algorithm that allows inverting a modified matrix via an existing \texttt{QLSP\_solver}, thereby directly approximating the solution vector instead of approximating the inverse of the coefficient matrix. By carefully choosing the step size $\eta$, the proposed algorithm can effectively precondition the linear system to mitigate the dependence on condition numbers that hindered the applicability of previous approaches.

翻译：求解线性方程组是一个基础性问题，但对于高维情形，经典算法的计算量可能非常巨大。现有的量子算法能够在问题维度上为量子线性系统问题（QLSP）实现指数级加速，但即使这样的理论优势也受限于系数矩阵的条件数。在本工作中，我们受经典邻近点算法（PPA）启发，提出了一种新的QLSP量子算法。我们提出的方法可视为一种元算法，它允许通过现有的 \texttt{QLSP\_solver} 对修正后的矩阵求逆，从而直接逼近解向量而非逼近系数矩阵的逆。通过精心选择步长 $\eta$，所提算法能够有效地对线性系统进行预处理，以减轻以往方法中制约其应用性的条件数依赖问题。