AI 中文总结
该研究证明分布式量子算法无法以概率1对环着色，提出了首个区分量子策略与非量子分布的下界技术，得出量子计算对环3着色无优势的结论。
AI 中文摘要
我们证明，在由匿名相同计算机构成的环中，任何以概率1找到3着色的分布式量子算法都必须是全局的，即需要Ω(n)次通信轮次。由此可知，量子计算与通信对该问题并无帮助。此前所有关于分布式图算法中量子优势的下界均使用与物理因果性相关的论证，但已知此类论证无法排除环3着色的快速量子优势。具体而言，任何基于因果性的论证都会排除有限依赖着色的存在，然而Holroyd与Liggett（2016）已证明此类着色确实存在。因此，为解决该问题，我们需要一种“真正量子”的下界技术，该技术能区分（1）不违反物理因果性的分布与（2）可通过量子策略实现的分布。我们提出了该领域首个此类下界技术：首先，证明1轮量子算法无法以概率1打破对称性；其次，提出一种理想的隐形传态策略，可将T轮量子3着色算法转化为1轮打破对称性的量子算法，同时保持成功概率为1。综合以上两点，该下界得证。
英文摘要
We prove that any distributed quantum algorithm that finds a $3$-coloring with probability $1$ in a cycle of anonymous identical computers has to be global, that is, it needs $Ω(n)$ communication rounds. It follows that quantum computation and communication does not help with this problem. All prior lower bounds on quantum advantage in distributed graph algorithms use arguments related to physical causality. However, it is known that such arguments cannot rule out fast quantum advantage for $3$-coloring cycles. In particular, any causality-based argument would rule out the existence of finitely dependent coloring, but Holroyd and Liggett (2016) showed that such colorings do exist. Hence to tackle this problem, we need a ``genuinely quantum'' lower-bound technique that can distinguish between (1) distributions that do not violate physical causality vs. (2) distributions that can be realized with a quantum strategy. We present the first such lower-bound technique in this context. First, we show that $1$-round quantum algorithms cannot break symmetry with probability $1$. Second, we present a wishful teleportation strategy that can be used to turn $T$-round quantum 3-coloring algorithms into $1$-round quantum algorithms breaking symmetry, while preserving success probability $1$. Put together, the lower bound follows.
Comments25 pages, 3 figures
