Recovering the Original Simplicity: Succinct and Exact Quantum Algorithm for the Welded Tree Problem

Abstract

This work revisits quantum algorithms for the well-known welded tree problem, proposing a succinct quantum algorithm based on the simple coined quantum walks. It iterates the naturally defined coined quantum walk operator for a classically precomputed number of iterations, and measures. The number of iterations is linear in the depth of the tree. The success probability of this procedure is inversely linear in the depth of the tree. Moreover, it is the same for all instances of the problem of a fixed size, therefore, we can use the exact quantum amplitude amplification subroutine to answer with probability 1. This gives an exponential speedup over any classical algorithm for the same problem. The significance of the results may be seen as follows. (i) Our algorithm is rather simple compared with the one in (Jeffery and Zur, STOC’2023), which not only breaks the stereotype that coined quantum walks can only achieve quadratic speedups over classical algorithms, but also demonstrates the power of the simplest quantum walk model. (ii) Our algorithm achieves certainty of success for the first time. Thus, it becomes one of the few examples that exhibit exponential separation between exact quantum and randomized query complexities.