  F-polynomials and g-vectors were defined by Fomin and Zelevinsky to give a
formula which expresses cluster variables in a cluster algebra in terms of the
initial cluster data. A quantum cluster algebra is a certain noncommutative
deformation of a cluster algebra. In this paper, we define and prove the
existence of analogous quantum F-polynomials for quantum cluster algebras. We
prove some properties of quantum F-polynomials. In particular, we give a
recurrence relation which can be used to compute them. Finally, we compute
quantum F-polynomials and g-vectors for a certain class of cluster variables,
which includes all cluster variables in type A quantum cluster algebras.
