发表机构
Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; TraverseQuantum Co., Ltd.; Tsinghua University（中国科学院软件研究所; 中国科学院大学; TraverseQuantum有限公司; 清华大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一种量子线性系统算法，在给定常数因子估计s时，同时达到对块编码预言机和态制备预言机的最优查询复杂度，并证明匹配的同一实例下界，实现联合最优。
AI 中文摘要
量子线性系统算法通过块编码预言机 $U_A$ 访问系数矩阵，并通过态制备预言机 $U_b$ 访问右端向量。这两个预言机的最优查询复杂度分别已知：对 $U_A$ 的查询为 $O(\kappa\log(1/\varepsilon))$ 次，对 $U_b$ 的查询为 $O(\kappa/s)$ 次，其中 $s=\alpha\\|A^{-1}b\\|$。单个算法能否同时达到这两个界仍然是一个未解决的问题。我们肯定地回答了这个问题。给定 $s$ 的常数因子估计，我们给出一个量子线性系统算法，该算法同时使用 $O\left(\kappa\log\frac1\varepsilon\right)$ 次对 $U_A$ 的查询和 $O\left({\kappa}/{s}\right)$ 次对 $U_b$ 的查询，将归一化解制备到误差 $\varepsilon$ 以内。该算法将态制备与精度细化分离，因此只有矩阵查询次数依赖于目标精度。我们还证明了一个匹配的同一实例最坏情况下的下界：对于每个量子线性系统算法和每个解范数尺度，都存在一个单一的 Hermitian 实例，在该实例上该算法需要 $\Omega(\kappa\log(1/\varepsilon))$ 次矩阵查询和 $\Omega(\kappa/s)$ 次态制备查询。因此，两个预言机复杂度可以同时优化，所得的界是联合最优的。
英文摘要
Quantum linear-system algorithms access the coefficient matrix through a block-encoding oracle $U_A$ and the right-hand side through a state-preparation oracle $U_b$. The optimal query complexities of these two oracles are known separately: $O(κ\log(1/\varepsilon))$ queries to $U_A$ and $O(κ/s)$ queries to $U_b$, where $s=α\|A^{-1}b\|$. Whether a single algorithm can achieve both bounds simultaneously has remained open. We resolve this question affirmatively. Given a constant-factor estimate of $s$, we give a quantum linear-system algorithm that prepares the normalized solution to error $\varepsilon$ using simultaneously $ O\left(κ\log\frac1\varepsilon\right)$ queries to $U_A$ and $O\left(κ/{s}\right)$ queries to $U_b$. The algorithm separates state preparation from precision refinement, so only the matrix-query count depends on the target accuracy. We also prove a matching same-instance worst-case lower bound: for every quantum linear-system algorithm and every solution-norm scale, there exists a single Hermitian instance on which that algorithm requires both $Ω(κ\log(1/\varepsilon))$ matrix queries and $Ω(κ/s)$ state-preparation queries. Thus the two oracle complexities can be optimized simultaneously, and the resulting bounds are jointly optimal.
Comments37 pages
