发表机构
Center on Frontiers of Computing Studies, Peking University; School of Computer Science, Peking University; School of Electronics Engineering and Computer Science, Peking University（前沿计算研究中心，北京大学; 计算机学院，北京大学; 电子工程学院与计算机科学系，北京大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
该研究针对高维网格上寻找塔斯基不动点问题，提出树过滤敌手方法，证明了量子查询下界，在部分参数范围优于经典下界，为建立量子复杂度下界提供了新途径。
AI 中文摘要
克纳斯托尔-塔斯基（Knaster-Tarski）不动点定理指出，完备格上的每个单调函数都存在不动点。除了在序理论中的基础作用外，该定理及其算法变体已在经济学、博弈论和编程语言等领域得到广泛应用。虽然寻找塔斯基不动点的查询复杂度在经典模型中已被广泛研究，但量子环境下的相关研究相对较少。我们利用非负谱敌手方法，证明了在[n]^k上寻找单调函数不动点的量子查询下界为Ω(k log n)。在两个极端情况n=2和k=1时，我们的量子下界分别与之前的经典下界Ω(k)和Ω(log n)匹配。对于n,k≥2，当n<k时，我们的下界优于之前的最佳经典下界；当n≥k时，与已知经典下界相比，我们的下界差距为log n / log k。为构建敌手矩阵，我们提出了树过滤敌手方法（Tree--Filtration Adversary Method）。该方法除了能得到我们的下界外，还为非负谱敌手方法提供了更清晰的组合解释。当问题的困难实例具有树状结构，且其直觉与经典决策树下界类似时，我们的方法为建立量子复杂度下界提供了一种有前景的途径。
英文摘要
The Knaster-Tarski fixed-point theorem states that every monotone function over a complete lattice has a fixed point. Beyond its fundamental role in order theory, the theorem and its algorithmic variants have found broad applications in areas such as economics, game theory, and programming languages. While the query complexity of finding a Tarski fixed point has been extensively studied in classical models, comparatively little is known in the quantum setting. We prove an $Ω(k\log n)$ quantum query lower bound for finding a fixed point of a monotone function on $[n]^k$, using the nonnegative spectral adversary method. In the two extremal regimes $n = 2$ and $k = 1$, our quantum lower bound matches the previous classical lower bounds $Ω(k)$ and $Ω(\log n)$, respectively. For $n, k\geq 2$, our bound improves the best previous classical lower bound when $n < k$ and is within a factor of $\log n / \log k$ compared to the known classical lower bound when $n \geq k$. To construct the adversary matrix, we develop the Tree--Filtration Adversary Method. Besides yielding our lower bound, the method offers a more transparent combinatorial interpretation of the nonnegative spectral adversary method. When the hard instances of a problem admit a tree-like organization and suggest an intuition analogous to classical decision-tree lower bounds, our method provide a promising approach to establishing quantum complexity lower bounds.
Comments35 pages, 5 figures
