Hardness of Pathfinding in a Welded Tree

2026-09-17

AI Summary

研究解决了量子算法在焊接树中寻找从入口到出口路径的效率问题，通过证明任何量子查询算法至少需要指数级查询次数，使用压缩置换oracle技术构建数据库来追踪算法学习和遗忘的图信息。

Abstract

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.

Problem

• quantum algorithm
• welded tree
• pathfinding

Innovation

• quantum query lower bound
• compressed permutation oracle
• welded tree