



01818nas a2200169   4500008004100000020002200041245005100063210005100114300001600165490000800181520131000189100002301499700002101522700002501543700002401568856005601592       2023                        eng d  a978-3-95977-263-100aQuantum Algorithms and the Power of Forgetting0 aQuantum Algorithms and the Power of Forgetting  a37:1--37:220 v2513 a<p>The so-called welded tree problem provides an example of a black-box problem that can be solved exponentially faster by a quantum walk than by any classical algorithm [Andrew M. Childs et al., 2003]. Given the name of a special entrance vertex, a quantum walk can find another distinguished exit vertex using polynomially many queries, though without finding any particular path from entrance to exit. It has been an open problem for twenty years whether there is an efficient quantum algorithm for finding such a path, or if the path-finding problem is hard even for quantum computers. We show that a natural class of efficient quantum algorithms provably cannot find a path from entrance to exit. Specifically, we consider algorithms that, within each branch of their superposition, always store a set of vertex labels that form a connected subgraph including the entrance, and that only provide these vertex labels as inputs to the oracle. While this does not rule out the possibility of a quantum algorithm that efficiently finds a path, it is unclear how an algorithm could benefit by deviating from this behavior. Our no-go result suggests that, for some problems, quantum algorithms must necessarily forget the path they take to reach a solution in order to outperform classical computation.</p>1 aChilds, Andrew, M.1 aCoudron, Matthew1 aGilani, Amin, Shiraz1 aKalai, Yael, Tauman  uhttps://drops.dagstuhl.de/opus/volltexte/2023/17540