  In this paper we study the functions that can be learned through the
polynomial interpolation quantum algorithm designed by Childs et al. This
algorithm was initially intended to find the coefficients of a multivariate
polynomial function defined on finite fields $\mathbb{F}_q$. We extend its
scope to vector inner product functions of the form
$\mathcal{O}_{\mathbf{s}}(\mathbf{v}) = \mathbf{s}\cdot\mathbf{v}$ where the
goal is to find the vector $\mathbf{s} \in \mathbb{F}_q^n$. We examine the
necessary conditions on the domain $\mathcal{V}$ of $\mathcal{O}_{\mathbf{s}}$
and prove that the algorithm is optimal for such functions. Furthermore, we
show that the success probability approaches 1 for large $q$ and large domain
order $|\mathcal{V}|.$ Finally, we provide a conservative formula for the
number of queries required to achieve this success probability.
