AI 中文总结
研究量子资源在分布式计算中是否能提供优势。通过颜色提升技术给出 3 色有根树的轮数下限，表明量子资源无优势；还证明 2 色偶圈在量子 - LOCAL 模型中所需轮数，显示量子算法不比经典算法节省轮数。
AI 中文摘要
在过去十年中，人们致力于理解量子资源是否能在分布式计算中提供优势，特别是能否帮助克服网络中的局部性约束。最近，Coiteux - Roy 等人表明量子资源对无根树的 3 着色没有帮助。本文证明量子资源对有根树的 3 着色也无优势，还给出 2 色偶圈所需轮数结果。
英文摘要
Significant effort has been devoted over the past decade to understanding whether quantum resources can provide advantages in distributed computing, and in particular whether they can help overcome locality constraints in networks, typically in Linial's LOCAL model. Recently, Coiteux-Roy~et~al.~(STOC 2024) showed that quantum resources do not help for 3-coloring \textit{unrooted} trees: in particular, their lower bound holds in the stronger \textit{non-signaling} model, which formalizes the principle of physical causality in distributed computing. The case of \textit{rooted} trees, however, was left open by their work. For rooted trees, the deterministic Cole-Vishkin algorithm 3-colors $n$-node trees in $O(\log^\star n)$ rounds, matching Linial's classical $Ω(\log^\star n)$ lower bound (FOCS 1987). In this paper, we show that any algorithm in quantum-LOCAL (without pre-shared entanglement) that properly 3-colors $n$-node rooted trees with probability at least ${1-O(1/\log n)}$ must perform $Ω(\log^\star n)$ rounds. That is, quantum resources provide no advantage for 3-coloring rooted trees. To get this result, we show a lower bound of $Ω(\log^\star Δ)$ for 3-coloring any $Δ$-ary tree with success probability at least $1-1/Δ$. The proof uses a \textit{color lifting} technique that bears similarity to Linial's original argument. We also show, as a separate result, that 2-coloring even-length $n$-node cycles with probability $1-O(1/n)$ requires $n/2-1$ rounds in the quantum-LOCAL model, even with pre-shared entangled states. This improves the previously known $\lceil (n-2)/4 \rceil$ lower bound of Gavoille, Kosowski, and Markiewicz (DISC 2009) by a factor of two, and shows that quantum algorithms cannot save even a single round over classical deterministic algorithms for 2-coloring even-length cycles.
