Quantum Algorithms for Class Group of a Number Field

Problem Definition

Associated with each number field is a finite abelian group called the class group. The order of the class group is called the class number. Computing the class number and the structure of the class group of a number field are among the main tasks in computational algebraic number theory [3].       

A number field F can be defined as a subfield of the complex numbers $\mathbb{C}$ which is generated over the rational numbers $\mathbb{Q}$ by an algebraic number, i. e. $F = \mathbb{Q} (\theta)$ where θ is the root of a polynomial with rational coefficients. The ring of integers $\mathcal{O}$ of F is the subset consisting of all elements that are roots of monic polynomials with integer coefficients. The ring $\mathcal{O} \subseteq F$ can be thought of as a generalization of $\mathbb{Z}$, the ring of integers in $\mathbb{Q}$. In particular, one can ask whether $\mathcal{O}$ is a principal ideal domain and whether elements in $\mathcal{O}$...