  In high-redshift ($z>5$) quasars, a large amount of dust ($\textstyle\sim
10^{8} \mathrm{M}_{\sun}$) has been observed. In order to explain the large
dust content, we focus on a possibility that grain growth by the accretion of
heavy elements is the dominant dust source. We adopt a chemical evolution model
applicable to nearby galaxies but utilize parameters adequate to high-$z$
quasars. It is assumed that metals and dust are predominantly ejected by Type
II supernovae (SNe). We have found that grain growth strongly depends on the
grain size distribution. If we simply use the size distribution of grains
ejected from SNe, grain growth is inefficient because of the lack of small
grains (i.e.\ small surface-to-volume ratio of the dust grains). However, if we
take small grain production by interstellar shattering into consideration,
grain growth is efficient enough to account for the rich dust abundance in
high-$z$ quasars. Our results not only confirm that grain growth is necessary
to explain the large amount of dust in high-$z$ quasars, but also demonstrate
that grain size distributions have a critical impact on grain growth.
