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.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.

quant-pharXiv:2609.26712Sep 23, 20260 citations

Edge-Based Eulerian Quantum Walks: Spectral and Topological Properties for Quantum Transport

Pavitra Bhargavi Allamaraju, Tia Tia

Jun 29, 20260 citations

Ω(2^{n/12})

queries are required to find a path in a height-

n

tree with constant success probability.