  Let $g_S$ be the Simanca metric on the blow-up $\tilde{\mathbb{C}}^2$ of
$\mathbb{C}^2$ at the origin. We show that $(\tilde{\mathbb{C}}^2,g_S)$ admits
a regular quantization. We use this fact to prove that all coefficients in the
Tian-Yau-Zelditch expansion for the Simanca metric vanish and that a dense
subset of $(\tilde{\mathbb{C}}^2, g_S)$ admits a Berezin quantization
