Sample complexity of hidden subgroup problem

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