发表机构
Xiangtan University（湘潭大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出基于薛定谔化线性组合的量子算法求解线性系统，通过两个LCHS实例实现最优矩阵查询复杂度，无需变时间振幅放大。
AI 中文摘要
量子线性系统算法（QLSAs）旨在特定条件下以指数级速度优于经典方法求解线性系统 $A\bb{x}=\bb{b}$。在本工作中，我们从基于常微分方程（ODE）的视角开发了求解线性代数方程的量子算法。受傅里叶方法中哈密顿模拟线性组合（LCHS）表示的启发，我们将解 $\bb{x}$ 表示为线性对流方程组解的线性组合，该方程组在傅里叶域中变为具有酉演化的薛定谔型方程。我们将此表示称为LC-薛定谔化。基于此结果，我们构建了一个基于LCHS的量子算法，包含两个LCHS实例：一个用于时间推进，一个用于数值积分。关键构造使用高斯平滑帽函数的导数，并在固定的辅助区间上恢复解。这允许截断时间独立于目标精度，并避免因选择单个网格点而导致的成功概率损失。周期化和显式傅里叶系数提供了相应的投影误差界。在所述预言机假设下，并给定解范数的常数因子估计，对选择算子的直接模拟和块预处理实现了最优矩阵查询复杂度 $\mathcal{O}(\kappa_A\log\frac1\varepsilon)$，而无需使用变时间振幅放大（VTAA）。对右侧制备预言机的查询也满足相同的上界。
英文摘要
Quantum linear systems algorithms (QLSAs) aim to solve linear systems $A\bb{x}=\bb{b}$ exponentially faster than classical methods under certain conditions. In this work, we develop quantum algorithms for solving linear algebraic equations from an ODE-based perspective. Inspired by the linear combination of Hamiltonian simulation (LCHS) representation in the Fourier approach \cite{Childs2017QLSA}, we express the solution $\bb{x}$ as a linear combination of solutions to a system of linear convection equations, which become Schrödinger-type equations with unitary evolutions in the Fourier domain. We refer to this representation as LC-Schrödingerization. Based on this result, we construct an LCHS-based quantum algorithm with two LCHS instances: one for time-marching and one for numerical integration. The key construction uses the derivative of a Gaussian-smoothed hat function and recovers the solution over a fixed auxiliary interval. This permits a truncation time independent of the target accuracy and avoids the loss in success probability from selecting a single grid point. Periodization and explicit Fourier coefficients provide the corresponding projection error bounds. Under the stated oracle assumptions and given a constant-factor estimate of the solution norm, direct simulation of the select operators and block preconditioning achieve the optimal matrix-query complexity $\mathcal{O}(κ_A\log\frac1\varepsilon)$ without using variable-time amplitude amplification (VTAA). The same upper bound holds for queries to the right-hand-side preparation oracle.
