Recovering the original simplicity: succinct and deterministic quantum algorithm for the welded tree problem

Abstract

This work revisits quantum algorithms for the well-known welded tree problem, proposing a very succinct quantum algorithm based on the simplest coined quantum walks. It simply iterates the naturally defined coined quantum walk operator for a predetermined time and finally measure, where the predetermined time can be efficiently computed on classical computers. Then, the algorithm returns the correct answer deterministically, and achieves exponential speedups over any classical algorithm. 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 theoretically achieves zero-error, which is not possible with existing methods. Thus, it becomes one of the few examples that exhibit exponential separation between deterministic (exact) quantum and randomized query complexities, which may also change people's perception that since quantum mechanics is inherently probabilistic, it impossible to have a deterministic quantum algorithm with exponential speedups for the weled tree problem.