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

TL;DR

This paper presents a polynomial-time quantum algorithm for identifying hidden parabolic subgroups in general linear groups over finite fields, advancing quantum solutions for algebraic group problems.

Contribution

It introduces the first efficient quantum algorithm for general parabolic subgroups in $GL_n(F_q)$ without prior parameter knowledge, extending previous special cases.

Findings

Algorithm runs in polynomial time in $ ext{log } q$ and $n$

Works without prior knowledge of subgroup parameters

Extends quantum subgroup finding to all parabolic subgroups

Abstract

In the theory of algebraic groups, parabolic subgroups form a crucial building block in the structural studies. In the case of general linear groups over 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 lower triangular matrices where the $i$th block is of size $n_i$. Our main result is a quantum algorithm of time polynomial in $\log q$ and $n$ for solving the hidden subgroup problem in $GL_n(F_q)$, when the hidden subgroup is promised to be a parabolic subgroup. Our algorithm works with no prior knowledge of the parameter of the hidden parabolic subgroup. Prior to this work, such an efficient 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), and for minimal parabolic subgroups (Borel subgroups), for the case when $q$ is not much smaller than $n$ (G. Ivanyos: Quantum Inf. Comput., Vol. 12, pp. 661-669).