AI 中文总结
本研究证明在焊接树中寻找入口到出口路径的量子查询复杂度下界为Ω(2^{n/24})，表明该问题对量子算法同样困难。
AI 中文摘要
从焊接树的入口出发，量子游走算法找到出口顶点的速度比任何经典算法都快指数级。然而，是否存在任何量子算法能够高效地找到从入口到出口的路径，一直是一个悬而未决的问题。我们通过证明寻找此类路径的指数级量子查询下界来回答这个问题。具体而言，对于高度为$n$的树，任何量子查询算法为了以恒定概率成功，至少需要$\Omega(2^{n/24})$次查询。我们的证明使用了压缩排列预言机技术来构建数据库，以追踪算法已学习并遗忘的图信息，并表明没有高效的量子算法能在这些记录中构建从入口到出口的路径。
英文摘要
Starting from the entrance of a welded tree, a quantum walk algorithm can find its exit vertex exponentially faster than any classical algorithm. However, it has been an open question whether any quantum algorithm is able to efficiently find a path from the entrance to the exit. We answer this by proving an exponential quantum query lower bound for finding such path. Specifically, for trees of height $n$, any quantum query algorithm requires at least $Ω(2^{n/24})$ queries in order to succeed with constant probability. Our proof uses the compressed permutation oracle technique in order to construct databases which track the graph information an algorithm has learned and forgotten, and show that no efficient quantum algorithm can build an entrance-to-exit path in these records.
Comments55 pages, 4 figures
