Numbers-on-Forehead通信中的指数级量子优势
Exponential Quantum Advantage in Numbers-on-Forehead Communication
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- Penn State University（宾夕法尼亚州立大学）
- University of Southern California（南加州大学）
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中文总结 AI 辅助
本文首次在一般交互式三方NOF模型中构造交错酉乘积问题，证明其仅需O(log n)量子通信但需Ω~(n^{1/32})随机通信，实现指数级量子优势。
中文摘要 AI 辅助
我们首次在一般交互式三方Numbers-on-Forehead（NOF）模型中，针对一个判定问题给出了指数级量子优势。此前的分离仅适用于受限协议，例如针对关系问题的单向通信。我们构造了一个显式的部分布尔函数——交错酉乘积问题，它仅需$O(\log n)$的NOF量子通信，但需要$\widetilde{\Omega}(n^{1/32})$的随机化通信。该函数基于Arunachalam、Girish和Lifshitz（TQC 2024）提出的两方酉乘积问题。主要技术障碍在于：差异法（discrepancy）作为NOF的标准下界方法，同样也会对量子通信给出下界。我们转而发展了一种基于正则性的论证方法，用于推导NOF的随机化下界，该方法借鉴了Kelley、Lovett和Meka的思路，并将Abboud、Fischer、Kelley、Lovett和Meka（STOC 2024）的正则性分解适配到柱面交集上。结合Arunachalam、Girish和Lifshitz的矩阵乘积估计，我们得到了所需的随机化下界。
英文摘要
We give the first exponential quantum advantage in the general interactive three-party Numbers-on-Forehead (NOF) model for a decision problem. Previous separations hold only for restricted protocols like one-way communication for a relation. We construct an explicit partial Boolean function, the Interleaved Unitary Product problem, that requires only $O(\log n)$ NOF quantum communication but $\widetildeΩ(n^{1/32})$ randomized communication. This function builds on the two-party unitary product problem of Arunachalam, Girish, and Lifshitz (TQC 2024). The main technical obstacle is that discrepancy, the standard lower-bound method for NOF, also lower-bounds quantum communication. We instead develop a regularity-based argument for randomized NOF lower bounds, building on the approach of Kelley, Lovett, and Meka and adapting the regularity decomposition of Abboud, Fischer, Kelley, Lovett, and Meka (STOC 2024) to cylinder intersections. Combined with matrix-product estimates of Arunachalam, Girish, and Lifshitz, this yields our randomized lower bound.
