  In order to investigate the origin of quasars, we estimate the bias factor
for low-luminosity quasars at high redshift for the first time. In this study,
we use the two-point angular cross-correlation function (CCF) for both
low-luminosity quasars at $-24<M_{\rm 1450}<-22$ and Lyman-break galaxies
(LBGs). Our sample consists of both 25 low-luminosity quasars (16 objects are
spectroscopically confirmed low-luminosity quasars) in the redshift range
$3.1<z<4.5$ and 835 color-selected LBGs with $z^{\prime}_{\rm LBG}<25.0$ at
$z\sim4$ in the COSMOS field. We have made our analysis for the following two
quasar samples; (1) the spectroscopic sample (the 16 quasars confirmed by
spectroscopy), and (2) the total sample (the 25 quasars including 9 quasars
with photometric redshifts). The bias factor for low-luminosity quasars at
$z\sim4$ is derived by utilizing the quasar-LBG CCF and the LBG
auto-correlation function. We then obtain the $86\%$ upper limits of the bias
factors for low-luminosity quasars, that are 5.63 and 10.50 for the total and
the spectroscopic samples, respectively. These bias factors correspond to the
typical dark matter halo masses, log $(M_{\rm DM}/(h^{-1}M_{\odot}))=$$12.7$
and $13.5$, respectively. This result is not inconsistent with the predicted
bias for quasars which is estimated by the major merger models.
