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
}
\{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
)
}
\{U^{p(n)}\}
where
p
p
is a polynomial
Z
→
Z
\mathbb {Z}\to \mathbb {Z}
and
U
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”
Z
≀
Z
\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 Følner averages of unitary polynomial actions of any abelian group
G
\mathbb {G}
in place of
Z
\mathbb {Z}
.