Efficient path finding in the original welded-tree distribution

Determine whether a quantum algorithm can efficiently find an explicit path between the entrance and exit roots in the original welded tree problem, where the leaves of two binary trees are joined by a uniformly random alternating cycle, rather than by the independently matched weld distribution analyzed here.

Background

The welded tree problem exhibits an exponential quantum speedup for reaching the exit root, but the known quantum walk does not output any individual entrance-to-exit path because recording path information destroys the interference responsible for the speedup. The paper proves an exponential query lower bound for the closely related independently matched welded-tree ensemble.

The original welded-tree construction uses a random cycle to join the leaves. The authors state that their argument should be adaptable to this original distribution, but the cited open problem concerns whether any quantum algorithm can efficiently find a path in that original setting; the paper does not establish such a result for the random-cycle ensemble.