焊接树中路径查找的量子下界
A quantum lower bound for path finding in welded trees
- University of Maryland（马里兰大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
该论文证明了在焊接树问题中，任何量子算法都需要指数级多的查询才能找到根之间的路径，尽管量子游走能指数级加速导航，但记录路径会破坏干涉，从而无法高效找到具体路径。
AI中文摘要：
在焊接树问题中，算法需要在一张由两棵二叉树在叶节点处通过连接边的“焊接”形成的图上进行导航。量子游走可以比任何经典算法指数级更快地从根节点导航到另一个根节点。然而，已知的高效量子算法无法找到根之间的路径，因为记录路径会破坏相长干涉，从而丧失加速优势。我们证明这是固有的：任何量子算法都需要指数级多的查询才能在独立匹配的焊接树图的根之间找到一条路径。这提供了一个问题的例子，即量子计算机可以通过叠加态指数级地探索大量路径，从而比任何经典算法指数级更快地解决该问题，但证明性地难以找到任何这样的路径。该证明使用压缩置换预言机来记录量子算法查询图时的进展。我们表明，压缩数据库在较小误差内保持无路径状态。通过控制这些误差并限制算法每次压缩预言机查询的进展，我们证明在高度为n的树中，以恒定成功概率找到一条路径需要Ω(2^{n/12})次查询。
英文摘要：
In the welded tree problem, an algorithm is tasked with navigating a graph formed from two binary trees joined at the leaves through a ``weld'' of connecting edges. A quantum walk can navigate from root to root exponentially faster than any classical algorithm. However, known efficient quantum algorithms cannot find a path between the roots, as recording the path destroys constructive interference and thus the speedup. We prove that this is inherent: any quantum algorithm needs exponentially many queries to find a path between the roots of an independently matched welded tree graph. This provides an example of a problem that a quantum computer can solve exponentially faster than any classical algorithm by exploring exponentially many paths in superposition, but where it is provably intractable to find any such path. The proof uses compressed permutation oracles to record the progress of a quantum algorithm as it queries the graph. We show that the compressed database remains path-free up to a small error. By controlling such errors and bounding the progress of the algorithm with each compressed oracle query, we show that $Ω(2^{n/12})$ queries are required to find a path in a height-$n$ tree with constant success probability.
