发表机构
IRIF (CNRS & Université Paris Cité)（IRIF（法国国家科学研究中心与巴黎西岱大学））
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究随机顺序对量子流处理的影响，提出补充概念，证明特定问题在随机顺序下量子算法高效而经典及顺序量子场景需多项式空间，并扩展多个下界与算法框架。
AI 中文摘要
随机顺序如何改变量子内存在流处理中的作用？后来的经典输入可以恢复先前查询所消耗的量子态的有用性。我们称之为补充。我们基于隐藏匹配构建了一个人工问题，具有重复的坐标数据和在线匹配请求。它在均匀随机顺序下允许使用多对数空间的单遍量子算法，但在随机顺序下经典地以及当所有更新先于请求时量子地，均无条件需要多项式空间。为了证明量子下界，我们加强了Gilboa、Jain和McClean关于多重隐藏匹配的可消耗性界限。在没有预先纠缠的情况下，任何支持$r$个独立匹配请求的量子编码需要$\Omega(r)$个量子比特，对于$r\le N^{1/2-\delta}$和每个固定的$\delta\in(0,1/2)$，即使同时揭示和任意联合解码也是如此。对于顺序请求，线性界限扩展到$r=\Theta(\sqrt N)$。我们还将Kallaugher的三角形计数算法适应于均匀随机流，其中每条边重复出现的次数相同。重建量子草图并重新采样经典估计器在适当的参数范围内改善了其期望空间界限。最后，我们将Assadi和Sundaresan的鲁棒噪声间隙循环框架扩展到量子流。块隐藏异或的量子通信下界在随机边顺序下为足够大的循环产生$\Omega(n)$空间下界，对几个图问题有影响。因此，随机顺序可以促进小型量子表示的补充，而其他任务仍需要大量空间。
英文摘要
How can random order change the role of quantum memory in streaming? Later classical input can restore the usefulness of a quantum state consumed by earlier queries. We call this replenishment. We construct an artificial problem based on Hidden Matching, with repeated coordinate data and online matching requests. It admits a one-pass quantum algorithm using polylogarithmic space in uniformly random order, but unconditionally requires polynomial space both classically in random order and quantumly when all updates precede the requests. To prove the quantum lower bound, we strengthen the consumability bounds of Gilboa, Jain, and McClean for Multiple Hidden Matching. Without prior entanglement, any quantum encoding supporting $r$ independent matching requests requires $Ω(r)$ qubits for $r\le N^{1/2-δ}$ and every fixed $δ\in(0,1/2)$, even with simultaneous revelation and arbitrary joint decoding. For sequential requests, the linear bound extends through $r=Θ(\sqrt N)$. We also adapt Kallaugher's triangle-counting algorithm to uniformly random streams in which every edge is repeated equally often. Rebuilding the quantum sketch and resampling the classical estimator improve its expected-space bound in suitable parameter regimes. Finally, we extend the robust Noisy Gap Cycle framework of Assadi and Sundaresan to quantum streaming. A quantum communication lower bound for Block Hidden XOR yields an $Ω(n)$ space lower bound in random edge order for large enough cycles, with consequences for several graph problems. Thus random order can enable replenishment of small quantum representations, while substantial space requirements persist for other tasks.
