量子线性系统问题的薛定谔化方法

中文摘要 AI 辅助

我们在两个更高维度上为量子线性系统问题开发了一种薛定谔化算法。此前的LC-薛定谔化方法将解表示为齐次对流响应的时间积分，并通过演化时间上的酉算子线性组合（LCU）来实现该积分。我们转而使用Duhamel原理将该积分纳入一个具有零初始数据的非齐次对流方程中。随后，在对流变量中进行傅里叶投影，并在第二个辅助变量中进行薛定谔化，从而无需单独的LCU步骤即可实现解的求解。核函数与恢复过程的联合选择使得演化时间与目标精度无关，$L^2$核归一化一致有界，并能在固定区间内实现精确恢复。我们建立了周期化和离散化误差界，并分析了区间恢复概率。通过块预处理，我们保留了单一的对数精度因子，并在线性条件数依赖下无需变时振幅放大即可实现。在精确预言机访问下，给定有效的常数因子解范数估计，该算法对每个原始输入预言机使用$\mathcal O(\kappa_A\log(1/\varepsilon))$次查询，具有常数成功概率，且$\ell^2$状态误差至多为$\varepsilon$。当所提供的范数界是紧的时，矩阵查询界与标准最坏情况缩放相匹配。UnitaryLab在正定和不定系统上的模拟展示了区间恢复带来的解精度和概率增益。

英文摘要

We develop a Schrödingerization algorithm for quantum linear systems problems in two higher dimensions. The earlier LC-Schrödingerization approach represents the solution as a time integral of a homogeneous convection response and implements this integral by a linear combination of unitaries (LCU) over evolution times. We instead use Duhamel's principle to incorporate the integral into an inhomogeneous convection equation with zero initial data. Fourier projection in the convection variable and Schrödingerization in a second auxiliary variable then realize the solution without the separate LCU step. A joint choice of the kernel and recovery procedure gives an evolution time independent of the target accuracy, a uniformly bounded $L^2$ kernel normalization, and accurate recovery on a fixed interval. We establish the periodization and discretization bounds and analyze the interval recovery probability. With block preconditioning, we retain a single logarithmic precision factor and obtain linear condition-number dependence without variable-time amplitude amplification. Under exact oracle access and given a valid constant-factor solution-norm estimate, the algorithm uses $\mathcal O(κ_A\log(1/\varepsilon))$ queries to each original input oracle, with constant success probability and $\ell^2$ state error at most $\varepsilon$. The matrix-query bound matches the standard worst-case scaling when the supplied norm bounds are tight. UnitaryLab simulations on positive-definite and indefinite systems demonstrate the solution accuracy and probability gain from interval recovery.

发表机构

• Xiangtan University（湘潭大学）

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