Global Phase Helps in Quantum Search: Yet Another Look at the Welded Tree Problem

TL;DR

This paper presents a simplified proof of the optimal linear hitting time for the welded tree problem using a discrete-time quantum walk, demonstrating the effectiveness of the electric quantum walk framework and its applicability to similar hierarchical graphs.

Contribution

It provides a concise proof of the welded tree problem's quantum speed-up using a modified electric quantum walk, extending the approach to other hierarchical graph structures.

Findings

Proves linear hitting time for welded tree problem with quantum walk

Introduces a simple modification to the electric quantum walk framework

Extends technique to other 1D hierarchical graphs

Abstract

Up to now, relatively few exponential quantum speed-ups have been achieved. Out of them, the welded tree problem (Childs, Cleve, Deotto, Farhi, Gutmann, and Spielman'2003) is one of the unusual examples, as the exponential speed-up is attained by a quantum walk. In this paper, we give a very short proof of the optimal linear hitting time for this problem by a discrete-time quantum walk, which is based on a simple modification of the electric quantum walk framework. The same technique can be applied to other 1-dimensional hierarchical graphs, yielding results similar to (Balasubramanian, Li, and Harrow'2023).