  Pseudoscalar-pole diagrams are an important component of estimates of the
hadronic light-by-light (HLbL) contribution to the muon $g-2$. We report on our
computation of the transition form factors $\mathcal{F}_{P \rightarrow \gamma^*
\gamma^*}$ for the neutral pseudoscalar mesons $P=\pi^0$ and $\eta$. The
calculation is performed using twisted-mass lattice QCD with physical quark
masses. On the lattice, we have access to a broad range of (space-like) photon
four-momenta and therefore produce form factor data complementary to the
experimentally accessible single-virtual direction, which directly leads to an
estimate of the pion- and $\eta$-pole components of the muon $g-2$. For the
pion, our result for the $g-2$ contribution in the continuum is comparable with
previous lattice and data-driven determinations, with combined relative
uncertainties below $10\%$. For the $\eta$ meson, we report on a preliminary
determination from a single lattice spacing.
