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In the theory of algebraic groups, parabolic subgroups form a crucialbuilding block in the structural studies. In the case of general linear groupsover a finite field $F_q$, given a sequence of positive integers $n_1, ...,n_k$, where $n=n_1+...+n_k$, a parabolic subgroup of parameter $(n_1, ...,n_k)$ in $GL_n(F_q)$ is a conjugate of the subgroup consisting of block lowertriangular matrices where the $i$th block is of size $n_i$. Our main result isa quantum algorithm of time polynomial in $\log q$ and $n$ for solving thehidden subgroup problem in $GL_n(F_q)$, when the hidden subgroup is promised tobe a parabolic subgroup. Our algorithm works with no prior knowledge of theparameter of the hidden parabolic subgroup. Prior to this work, such anefficient quantum algorithm was only known for the case $n=2$ (A. Denney, C.Moore, and A. Russell (2010), Quantum Inf. Comput., Vol. 10, pp. 282-291), andfor minimal parabolic subgroups (Borel subgroups), for the case when $q$ is notmuch smaller than $n$ (G. Ivanyos: Quantum Inf. Comput., Vol. 12, pp. 661-669).

An efficient quantum algorithm for finding hidden parabolic subgroups in the general linear group