Decomposing Finite Abelian Groups

TL;DR

This paper introduces a quantum algorithm for efficiently decomposing finite Abelian groups, which is crucial for the Abelian hidden subgroup problem and has implications for computing class numbers under the Generalized Riemann Hypothesis.

Contribution

It presents a novel quantum algorithm that improves the efficiency of decomposing finite Abelian groups, enabling applications in hidden subgroup problems and class number computations.

Findings

Quantum algorithm for Abelian group decomposition

Efficient computation of class numbers under GRH

Implications for quantum algorithms in number theory

Abstract

This paper describes a quantum algorithm for efficiently decomposing finite Abelian groups. Such a decomposition is needed in order to apply the Abelian hidden subgroup algorithm. Such a decomposition (assuming the Generalized Riemann Hypothesis) also leads to an efficient algorithm for computing class numbers (known to be at least as difficult as factoring).