Deterministic algorithms for the hidden subgroup problem

Abstract

The hidden subgroup problem (HSP) plays a crucial role in the field of quantum computing, since several celebrated quantum algorithms including Shor's algorithm have a uniform description in the framework of HSP. The problem is as follows: for a finite group G and a finite set X, given a function f:G→X and the promise that for any g1,g2∈G,f(g1)=f(g2) iff g1H=g2H for a subgroup H≤G, the goal of the decision version is to determine whether H is trivial, and the goal of the search version is to find H. Nayak (2021) asked whether there exist deterministic algorithms with O(|G||H|) query complexity for HSP. We answer this problem for Abelian groups, which also extends the main results of Ye et al. (2021), since here the algorithms do not rely on any prior knowledge of H.