# generalizations of hidden subgroup problem

Quantum Fourier Sampling tries to solve hidden subgroup problem which is defined via a map $$f$$ from group $$\mathrm{G}$$ to some set $$X$$ that separates cosets of sum unknown subgroup $$\mathrm{H}$$.

$$f(g_1)=f(g_2)$$ iff $$g_1 \in g_2 \mathrm{H}$$. The task is to find (generators of) $$\mathrm{H}$$. The underlying map $$f$$ partitions $$G$$ into cosets of $$\mathrm{H}$$. Are there any generalizations where $$f$$ is not necessarily distinct on distinct cosets. In particular if there are some cosets which are 1 and some others are 0, and we ask to find either $$\mathrm{H}$$, how effective is QFS? If not, then how close can we go? Lets say finding some or all cosets of $$H$$ going to 0.