随机数据流中的指数级量子空间优势
Exponential quantum space advantage in random data streams
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中文总结 AI 辅助
本研究证明随机顺序流模型中两个无条件的量子空间优势：Yamakawa--Zhandry码交集问题具有指数级量子优势，且最优多项式交集问题在低速率参数范围内仅需O(d log n)量子比特而经典需Ω(n)比特。
中文摘要 AI 辅助
我们在随机顺序流模型中展示了两个无条件的量子空间优势。首先，我们证明了Yamakawa--Zhandry码交集问题在其输入以随机顺序流式传输时，在流模型中具有指数级的量子优势。这意味着即使量子计算机仅以均匀随机顺序接收均匀随机函数$f$的$(x,f(x))$对，它们也展现出指数级的空间优势。我们的下界是通过使用Göös、Gur、Jain和Li（STOC 2025）工作中的密度恢复划分并结合凸势函数来证明的，类似于Raz（2018年，此http URL）关于奇偶性学习的工作以及Garg、Raz和Tal（STOC 2018）对其的推广。其次，我们使用我们的框架，通过解码量子干涉测量算法（Nature 2025；arXiv:2510.10967）的流式版本，展示了在某些参数范围内最优多项式交集（OPI）问题的量子空间优势。特别地，我们证明对于次数$d$和$n$个评估点，通过流式处理获得$1/2 + \Omega(\sqrt{d/n})$比例的满足OPI约束需要$\Omega(n)$经典比特的内存，但仅需$O(d\log n)$量子比特。这就在评估点数量远大于次数的“低速率”参数范围内产生了可证明的量子优势，且在某些参数设置下，空间优势可以大到指数级。
英文摘要
We show two unconditional quantum space advantages in the random-order streaming model. First, we show the Yamakawa--Zhandry Code Intersection problem admits exponential quantum advantage in the streaming model when its inputs are streamed in random order. This means quantum computers exhibit exponential space advantage even when simply receiving $(x,f(x))$ pairs for a uniformly random function $f$ in a uniformly random order. Our lower bound is shown using density-restoring partitions as in the work of Göös, Gur, Jain, and Li (STOC 2025) combined with a convex potential, similar to the work of Raz (J.ACM 2018) on parity learning and its generalization by Garg, Raz, and Tal (STOC 2018). Second, we use our framework to show quantum space advantage for the Optimal Polynomial Intersection (OPI) problem in certain regimes via a streaming version of the Decoded Quantum Interferometry algorithm (Nature 2025; arXiv:2510.10967). In particular, we show that for degree $d$ and $n$ evaluation points, attaining $1/2 + Ω(\sqrt{d/n})$ fraction of satisfied OPI constraints via streaming requires $Ω(n)$ classical bits of memory but only $O(d\log n)$ qubits. This yields provable quantum advantage in a "low-rate" regime when the number of evaluation points is much larger than the degree, with a space advantage that can be as large as exponential in certain parameter settings.
发表机构
- Stanford University（斯坦福大学）
- The University of Texas at Austin（德克萨斯大学奥斯汀分校）
机构由 AI 辅助整理，请以论文原文为准。
