发表机构
Virginia Tech（弗吉尼亚理工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本研究证明稀疏量子线性系统求解器在稀疏访问模型中的查询复杂度下界为 $\Omega(\kappa\sqrt{s}\log(1/\varepsilon))$，同时确立对条件数、稀疏度和精度的最优依赖，完善了该领域的理论图景。
AI 中文摘要
量子线性系统求解器是量子计算中的核心算法原语之一，其应用涵盖微分方程、优化以及机器学习等领域。其成本通常通过查询复杂度来衡量，即访问输入矩阵所需的预言机调用次数。在稀疏访问模型中，该复杂度由三个参数决定：条件数 $\kappa$、输入矩阵的稀疏度 $s$ 以及目标精度 $\varepsilon$。对 $\kappa$ 和 $\varepsilon$ 的依赖已通过下界 $\Omega(\kappa\log(1/\varepsilon))$ 得到充分理解，该下界与这些参数的最佳已知缩放相匹配。然而，一旦包含稀疏度 $s$，预期的下界长期以来被猜想为 $\Omega(\kappa\sqrt{s}\log(1/\varepsilon))$。最近 Mori 等人 [Quantum Sci. Tech. 11 035063 (2026)] 的工作通过建立常数误差 $\varepsilon$ 下的下界 $\Omega(\kappa\sqrt{s})$，朝这一目标迈出了重要一步。在本工作中，我们完善了这一图景，在稀疏访问模型中证明了完整的联合下界 $\Omega(\kappa\sqrt{s}\log(1/\varepsilon))$，从而同时确立了所有三个参数的预期依赖关系。
英文摘要
Quantum linear system solvers form one of the central algorithmic primitives in quantum computing, with applications ranging from differential equations and optimization to machine learning. Their cost is commonly measured through query complexity, which counts the number of oracle calls needed to access the input matrix. In the sparse-access model, this complexity is governed by three parameters: the condition number $κ$, the sparsity $s$ of the input matrix, and the target precision $\varepsilon$. The dependence on $κ$ and $\varepsilon$ is already well understood through the lower bound $Ω(κ\log(1/\varepsilon))$, which matches the best known scaling in these parameters. Once the sparsity $s$ is included, however, the expected lower bound has long been conjectured to be $Ω(κ\sqrt{s}\log(1/\varepsilon))$. Recent work by Mori et al. [Quantum Sci. Tech. 11 035063 (2026)] made an important step towards this goal by establishing the lower bound $Ω(κ\sqrt{s})$ for constant error $\varepsilon$. In this work, we complete the picture and prove the full joint lower bound $Ω(κ\sqrt{s}\log(1/\varepsilon))$ in the sparse-access model, thereby establishing the anticipated dependence on all three parameters simultaneously.
Comments21 pages
