A quantum polylog algorithm for non-normal maximal cyclic hidden subgroups in the affine group of a finite field

TL;DR

This paper presents a quantum algorithm that efficiently solves the hidden subgroup problem for certain non-normal cyclic subgroups in the affine group over finite fields, achieving polylogarithmic complexity.

Contribution

It introduces a novel quantum algorithm specifically designed for non-normal maximal cyclic subgroups in the affine group, improving computational efficiency.

Findings

Algorithm solves the problem with high probability (1 - ε)

Complexity is polylogarithmic in the size of the finite field

Applicable to affine groups of finite fields

Abstract

We give an algorithm to solve the quantum hidden subgroup problem for maximal cyclic non-normal subgroups of the affine group of a finite field (if the field has order $q$ then the group has order $q(q-1)$) with probability $1-\varepsilon$ with (polylog) complexity $O(\log(q)^{R}\log(\varepsilon)^{2})$ where $R<\infty.$