  The entropy production prior to BBN era is one of ways to prevent QCD axion
with the decay constant $F_{a}\in[10^{12}{\rm GeV},10^{16}{\rm GeV}]$ from
overclosing the universe when the misalignment angle is $\theta_{\rm
i}=\mathcal{O}(1)$. As such, it is necessarily accompanied by an early
matter-dominated era (EMD) provided the entropy production is achieved via the
decay of a heavy particle. In this work, we consider the possibility of
formation of primordial black holes during the EMD era with the assumption of
the enhanced primordial scalar perturbation on small scales ($k>10^{4}{\rm
Mpc}^{-1}$). In such a scenario, it is expected that PBHs with axion halo
accretion develop to ultracompact minihalos (UCMHs). We study how UCMHs so
obtained could be of great use in the experimental search for QCD axion dark
matter with $F_{a}\in[10^{12}{\rm GeV},10^{16}{\rm GeV}]$.

High Energy Physics - Phenomenology;Cosmology and Nongalactic Astrophysics;High Energy Astrophysical Phenomena