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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.

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