Finding hidden Borel subgroups of the general linear group

TL;DR

This paper introduces a quantum algorithm to efficiently identify hidden Borel subgroups within the general linear group over finite fields, advancing quantum solutions for algebraic group problems.

Contribution

It presents a polynomial-time quantum algorithm for the hidden subgroup problem in the general linear group, specifically targeting conjugates of invertible lower triangular matrices.

Findings

Quantum algorithm solves the hidden subgroup problem efficiently.

Algorithm's complexity is polynomial when the field size is comparable to the matrix degree.

Advances quantum approaches to algebraic group problems.

Abstract

We present a quantum algorithm for solving the hidden subgroup problem in the general linear group over a finite field where the hidden subgroup is promised to be a conjugate of the group of the invertible lower triangular matrices. The complexity of the algorithm is polynomial when size of the base field is not much smaller than the degree.