Sample complexity of hidden subgroup problem

Abstract

The hidden subgroup problem (HSP) has been attracting much attention in quantum computing, since several well-known quantum algorithms including Shor's algorithm can be described in a uniform framework as quantum methods to address different instances of it. One of the central issues about HSP is to characterize its quantum/classical complexity. For example, from the viewpoint of learning theory, sample complexity is a crucial concept. However, while the quantum sample complexity of the problem has been studied, a full characterization of the classical sample complexity of HSP seems to be absent, which will thus be the topic in this paper. HSP over a finite group is defined as follows: For a finite group G and a finite set V, given a function f:G→V and the promise that for any x,y∈G,f(x)=f(xy) iff y∈H for a subgroup H∈H, where H is a set of candidate subgroups of G, the goal is to identify H. Our contributions are as follows:i)For HSP, we show that the number of uniform examples necessary to learn the hidden subgroup with bounded error is at least Ω(minH∈H′⁡max⁡{log⁡|H′|log⁡|G||H|,|G||H|log⁡|H′|log⁡|G||H|}), where H′=H∖{G}; on the other hand, O(maxH∈H⁡{r(H),|G||H|r(H)}) uniform examples are sufficient, where r(H) is the rank of H.ii)By concretizing the parameters of HSP, we consider a class of restricted Abelian hidden subgroup problem (rAHSP) and obtain the upper and lower bounds for the sample complexity of rAHSP.iii)We continue to discuss a special case of rAHSP, generalized Simon's problem (GSP), and show that the sample complexity of GSP is Θ(max⁡{k,k⋅pn−k}). Thus we obtain a complete characterization of the sample complexity of GSP.