  Let $\widetilde{\mathbb{Q}_p}$ be the field of $p$-adic numbers in the
language of rings. In this paper we consider the theory of
$\widetilde{\mathbb{Q}_p}$ expanded by two predicates interpreted by
multiplicative subgroups $\alpha^\mathbb{Z}$ and $\beta^\mathbb{Z}$ where
$\alpha, \beta\in\mathbb{N}$ are multiplicatively independent. We show that the
theory of this structure interprets Peano arithmetic if $\alpha$ and $\beta$
have positive $p$-adic valuation. If either $\alpha$ or $\beta$ has zero
valuation we show that the theory of $(\widetilde{\mathbb{Q}_p},
\alpha^\mathbb{Z}, \beta^\mathbb{Z})$ does not interpret Peano arithmetic. In
that case we also prove that the theory is decidable iff the theory of
$(\widetilde{\mathbb{Q}_p}, \alpha^\mathbb{Z}\cdot \beta^\mathbb{Z})$ is
decidable.
