发表机构
Institute of Software, Chinese Academy of Sciences; University of Chinese Academy of Sciences; University of Southern California; State Key Laboratory for Novel Software Technology, Nanjing University（中国科学院软件研究所; 中国科学院大学; 南加州大学; 南京大学新型软件技术国家重点实验室）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
研究单向量子通信中搜索问题的下界，开发基于矩阵差异的新方法，为碰撞查找和三角形查找建立紧密量子下界，恢复经典下界，避免布尔隐藏匹配约简失效。
AI 中文摘要
我们研究单向量子通信中搜索问题的下界。与决策问题不同，搜索问题可能有多个有效输出，这给标准量子下界技术带来根本障碍。我们通过开发一种基于矩阵差异的新方法克服了这一障碍，该方法使我们能够联合界定量子协议的输出测量。作为该方法的应用，我们在一些自然参数范围内为两个基本搜索问题建立了首个紧密量子下界：碰撞查找和三角形查找。对于碰撞查找，我们证明了紧密的\(\Omega(N^{1/4})\)单向量子通信下界。此前，由于Göös和Jain（RANDOM 2022），碰撞查找的最佳已知量子通信下界为\(\Omega(N^{1/12})\)，即使在单向限制下也没有更强的界。对于图流中的三角形查找，在\(1\le \Delta_V\le m^{2/3}\)条件下，我们证明了对于具有\(m\)条边、\(\Theta(m)\)个三角形且\(\Delta_E\)为常数的图，单通道量子流空间下界为\(\Omega\left(\sqrt{\Delta_V}\right)\)，这构成了该范围内首个非平凡量子空间下界，与Jayaram和Kallaugher（RANDOM 2021）的经典上界在对数因子上匹配。值得注意的是，我们的方法还通过完全不同的论证恢复了Kallaugher和Price（SODA 2017）的经典下界，避免了他们在量子协议中失效的布尔隐藏匹配约简。
英文摘要
We study one-way quantum communication lower bounds for search problems. Unlike decision problems, search problems can have many valid outputs, which pose a fundamental barrier to standard quantum lower-bound techniques. We overcome this by developing a novel method based on matrix discrepancy, which allows us to bound the output measurements of a quantum protocol jointly. As applications of our method, we establish the first tight quantum lower bounds for two fundamental search problems in some natural parameter regimes: collision finding and triangle finding. For collision finding, we prove a tight $Ω(N^{1/4})$ one-way quantum communication lower bound. Previously, the best-known quantum communication lower bound for collision finding was $Ω(N^{1/12})$ due to Göös and Jain (RANDOM 2022), and no stronger bound was known even under the one-way restriction. For triangle finding in graph streams, we prove a one-pass quantum streaming space lower bound of $Ω\left(\sqrt{Δ_V}\right)$ for graphs with $m$ edges, $Θ(m)$ triangles, and constant $Δ_E$, where $Δ_V$ and $Δ_E$ denote the maximum number of triangles sharing a common vertex and edge, respectively, under the condition that $1\le Δ_V\le m^{2/3}$. This constitutes the first nontrivial quantum space lower bound in this regime, matching the classical upper bound of Jayaram and Kallaugher (RANDOM 2021) up to logarithmic factors. Notably, our method also recovers the classical lower bound of Kallaugher and Price (SODA 2017) through an entirely different argument, avoiding their Boolean-Hidden-Matching reduction that breaks down for quantum protocols.
