Given a matrix $A$ of dimension $M \times N$ and a vector $\vec{b}$, the quantum linear system (QLS) problem asks for the preparation of a quantum state $|\vec{y}\rangle$ proportional to the solution of $A\vec{y} = \vec{b}$. Existing QLS algorithms have runtimes that scale linearly with the condition number $\kappa(A)$, the sparsity of $A$, and logarithmically with inverse precision, but often overlook structural properties of $\vec{b}$, whose alignment with $A$'s eigenspaces can greatly affect performance. In this work, we present a new QLS algorithm that explicitly leverages the structure of the right-hand side vector $\vec{b}$. The runtime of our algorithm depends polynomially on the sparsity of the augmented matrix $H = [A, -\vec{b}]$, the inverse precision, the $\ell_2$ norm of the solution $\vec{y} = A^+ \vec{b}$, and a new instance-dependent parameter \[ ET= \sum_{i=1}^M p_i^2 \cdot d_i, \] where $\vec{p} = (AA^{\top})^+ \vec{b}$, and $d_i$ denotes the squared $\ell_2$ norm of the $i$-th row of $H$. We also introduce a structure-aware rescaling technique tailored to the solution $\vec{y} = A^+ \vec{b}$. Unlike left preconditioning methods, which transform the linear system to $DA\vec{y} = D\vec{b}$, our approach applies a right rescaling matrix, reformulating the linear system as $AD\vec{z} = \vec{b}$. As an application of our instance-aware QLS algorithm and new rescaling scheme, we develop a quantum algorithm for solving multivariate polynomial systems in regimes where prior QLS-based methods fail. This yields an end-to-end framework applicable to a broad class of problems. In particular, we apply it to the maximum independent set (MIS) problem, formulated as a special case of a polynomial system, and show through detailed analysis that, under certain conditions, our quantum algorithm for MIS runs in polynomial time.

翻译：给定一个维度为 $M \times N$ 的矩阵 $A$ 和一个向量 $\vec{b}$，量子线性系统（QLS）问题要求制备一个与 $A\vec{y} = \vec{b}$ 的解成正比的量子态 $|\vec{y}\rangle$。现有的 QLS 算法其运行时间与条件数 $\kappa(A)$、矩阵 $A$ 的稀疏度呈线性关系，并与逆精度呈对数关系，但常常忽略了向量 $\vec{b}$ 的结构特性，而 $\vec{b}$ 与 $A$ 的特征子空间的对齐情况会极大地影响算法性能。在本工作中，我们提出了一种新的 QLS 算法，该算法明确利用了右侧向量 $\vec{b}$ 的结构。我们算法的运行时间多项式依赖于增广矩阵 $H = [A, -\vec{b}]$ 的稀疏度、逆精度、解 $\vec{y} = A^+ \vec{b}$ 的 $\ell_2$ 范数，以及一个新的实例相关参数 \[ ET= \sum_{i=1}^M p_i^2 \cdot d_i, \] 其中 $\vec{p} = (AA^{\top})^+ \vec{b}$，$d_i$ 表示 $H$ 的第 $i$ 行的 $\ell_2$ 范数的平方。我们还引入了一种针对解 $\vec{y} = A^+ \vec{b}$ 量身定制的结构感知重缩放技术。与将线性系统变换为 $DA\vec{y} = D\vec{b}$ 的左预处理方法不同，我们的方法应用右重缩放矩阵，将线性系统重新表述为 $AD\vec{z} = \vec{b}$。作为我们实例感知的 QLS 算法和新重缩放方案的应用，我们开发了一种用于求解多元多项式系统的量子算法，适用于先前基于 QLS 的方法失效的体系。这产生了一个适用于广泛问题类别的端到端框架。特别地，我们将其应用于最大独立集（MIS）问题，该问题被表述为多项式系统的一个特例，并通过详细分析表明，在特定条件下，我们用于 MIS 的量子算法在多项式时间内运行。