  We use model theory of metric structures to prove the pointwise convergence,
with a uniform metastability rate, of averages of a polynomial sequence
$\{T_n\}$ (in Leibman's sense) of unitary transformations of a Hilbert space.
As a special case, this applies to unitary sequences $\{U^{p(n)}\}$ where $p$
is a polynomial $\mathbb{Z}\to\mathbb{Z}$ and $U$ a fixed unitary operator;
however, our convergence results hold for arbitrary Leibman sequences. As a
case study, we show that the non-nilpotent "lamplighter group"
$\mathbb{Z}\wr\mathbb{Z}$ is realized as the range of a suitable quadratic
Leibman sequence. We also indicate how these convergence results generalize to
arbitrary Folner averages of unitary polynomial actions of any abelian group
$\mathbb{G}$ in place of $\mathbb{Z}$.
