  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.
