Efficient computations in central simple algebras using Amitsur cohomology

TL;DR

This paper introduces a quantum algorithm for explicitly determining isomorphisms of central simple algebras over number fields, leveraging Brauer factor sets and polynomial quantum algorithms for number theory tasks.

Contribution

It provides a novel quantum algorithm for the explicit isomorphism problem in central simple algebras, utilizing Amitsur cohomology and heuristic assumptions.

Findings

Quantum algorithm runs in polynomial time under heuristic assumptions.

Algorithm works efficiently for bounded degree algebras without heuristics.

Improves computational methods for central simple algebras over number fields.

Abstract

We present an efficient computational representation of central simple algebras using Brauer factor sets. Using this representation and polynomial quantum algorithms for number theoretical tasks such as factoring and $S$-unit group computation, we give a polynomial quantum algorithm for the explicit isomorphism problem over number field, which relies on a heuristic concerning the irreducibility of the characteristic polynomial of a random matrix with algebraic integer coefficients. We present another version of the algorithm which does not need any heuristic but which is only polynomial if the degree of the input algebra is bounded.