  Let $U^N = (U_1^N,\dots, U^N_p)$ be a d-tuple of $N\times N$ independent Haar
unitary matrices and $Z^{NM}$ be any family of deterministic matrices in
$\mathbb{M}_N(\mathbb{C})\otimes \mathbb{M}_M(\mathbb{C})$. Let $P$ be a
self-adjoint non-commutative polynomial. In 1998, Voiculescu showed that the
empirical measure of the eigenvalues of this polynomial evaluated in Haar
unitary matrices and deterministic matrices converges towards a deterministic
measure defined thanks to free probability theory. Let now $f$ be a smooth
function, the main technical result of this paper is a precise bound of the
difference between the expectation of $$ \frac{1}{MN} \text{Tr}\left(
f(P(U^N\otimes I_M,Z^{NM})) \right) , $$ and its limit when $N$ goes to
infinity. If $f$ is seven times differentiable, we show that it is bounded by
$M^2 \left\Vert f\right\Vert_{\mathcal{C}^7} N^{-2}$. As a corollary we obtain
a new proof with quantitative bounds of a result of Collins and Male which
gives sufficient conditions for the operator norm of a polynomial evaluated in
Haar unitary matrices and deterministic matrices to converge almost surely
towards its free limit. Actually we show that if $U^N$ and $Y^{M_N}$ are
independent and $M_N = o(N^{1/3})$, then almost surely, the norm of any
polynomial in $(U^N\otimes I_{M_N}, I_N\otimes Y^{M_N})$ converges almost
surely towards its free limit.
