Hidden parabolic subgroups of ${\mathrm{G}\mathrm{L}}_n$ over finite chain rings

We give a bounded-error quantum polynomial-time algorithm for finding hidden parabolic subgroups of ${\mathrm{G}\mathrm{L}}_n(R)$ over any effectively presented finite chain ring $R$. We call auxiliary group elements or subgroups used to extract information through oracle queries \emph{probes}. For chain length at least two, elementary abelian socle probes first recover the complete flag modulo the maximal ideal. When the residue field has size $q\ge 3$, we reconstruct the flag recursively, lifting rank-one terminal members one digit at a time. The intermediate digits reduce to abelian hidden subgroup problems, while the final digit is a finite-field multiple-shift problem solved using learning from subset and Fourier sampling. For $q=2$, where distinct flags can have the same stabilizer, we instead lift the conjugate parabolic subgroup through the congruence filtration. A final linear system determines it modulo normalizer directions. The algorithm outputs generators with probability at least $1-\epsilon$ in time $\mathrm{p}\mathrm{o}\mathrm{l}\mathrm{y}(n,\log \mathrm{\mid }R\mathrm{\mid },\log (1\mathrm{/}\epsilon ))$. Consequences include ${\mathrm{G}\mathrm{L}}_n(\mathbb{Z}\mathrm{/}N\mathbb{Z})$ for every $N\ge 2$ and, with an explicit maximal-order presentation, ${\mathrm{G}\mathrm{L}}_n({\mathcal{O}}_{\mathcal{K}}\mathrm{/}N{\mathcal{O}}_{\mathcal{K}})$ for number fields $K$.