Quantum algorithms and the power of forgetting

TL;DR

This paper demonstrates that certain quantum algorithms cannot efficiently find paths in the welded tree problem, implying that quantum advantage may require algorithms to forget information during computation.

Contribution

It proves a no-go theorem for a class of quantum algorithms solving the welded tree path-finding problem, highlighting the necessity of forgetting information for quantum speedup.

Findings

Quantum algorithms in this class cannot find paths efficiently.

Forgetting information may be essential for quantum advantage.

The result clarifies limitations of quantum algorithms on specific black-box problems.

Abstract

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. 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.