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 $zsim4$ 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 $zsim4$ 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.
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