AI 中文总结
针对构造性对角拉姆齐问题，提出量子查询算法，以次线性查询复杂度找到并验证齐次集合，并给出量子下界及随机图上的多项式加速。
AI 中文摘要
构造性对角拉姆齐问题要求，在给定一个$N$顶点图的邻接预言机访问权限时，找出拉姆齐定理所保证阶数的团或独立集。我们给出一个有界误差量子算法，对于每个$K\ge2$和$N\ge4^{K-1}$，该算法使用$O\\!\left(2^K K\log\frac K\eta\right)$次边查询，以不超过$\eta$的失败概率找到并验证一个齐次$K$-集合。在拉姆齐尺度$N=2^n$下，这产生一个阶为$\lfloor n/2\rfloor+1$的齐次集合，使用$O(\sqrt N\log N\log(\log N/\eta))$次查询，改进了显式经典递归的$O(N)$次查询，并且据我们所知，给出了拉姆齐关系的第一个次线性最坏情况算法。我们还通过从碰撞寻找的归约推导出$\Omega(N^{1/12})$的量子下界。该算法在隐式候选集合上运行构造性递归。每个集合由邻接约束的短合取表示，并使用封顶未知解量子搜索进行采样，同时一个尺度感知的浓度调度在估计精度与采样更深集合的递增成本之间取得平衡。我们用$\Omega(N^{1-1/\sqrt2})$的随机化下界补充上界，该下界从在线拉姆齐数的随机画家分析中转移而来，在均匀分布$G(N,1/2)$上成立。在该分布上，贪心量子搜索仅使用$\widetilde O(N^{1/4})$次查询，为随机图上的拉姆齐搜索提供了可证明的多项式量子加速。我们还给出一个免估计的规模偏置递归，并将其扩展到任意固定数量的边颜色。
英文摘要
The constructive diagonal Ramsey problem asks, given adjacency-oracle access to an $N$-vertex graph, for a clique or independent set of the order guaranteed by Ramsey's theorem. We give a bounded-error quantum algorithm that, for every $K\ge2$ and $N\ge4^{K-1}$, finds and verifies a homogeneous $K$-set using $O\!\left(2^K K\log\frac Kη\right)$ edge queries with failure probability at most $η$. At the Ramsey scale $N=2^n$, this yields a homogeneous set of order $\lfloor n/2\rfloor+1$ using $O(\sqrt N\log N\log(\log N/η))$ queries, improving on the $O(N)$ queries of the explicit classical recursion and giving, to our knowledge, the first sublinear worst-case algorithm for the Ramsey relation. We also derive an $Ω(N^{1/12})$ quantum lower bound by a reduction from collision finding. The algorithm runs the constructive recursion over implicit candidate sets. Each set is represented by a short conjunction of adjacency constraints and sampled using capped unknown-solution quantum search, and a scale-aware concentration schedule balances estimation accuracy against the increasing cost of sampling deeper sets. We complement the upper bound with an $Ω(N^{1-1/\sqrt2})$ randomized lower bound, transported from the random-Painter analysis of online Ramsey numbers, which holds on the uniform distribution $G(N,1/2)$. On that distribution a greedy quantum search uses only $\widetilde O(N^{1/4})$ queries, giving a provable polynomial quantum speedup for Ramsey search on random graphs. We also give an estimation-free size-biased recursion and extend it to every fixed number of edge colours.
Comments19 pages
