<title>Abstract</title>
We present ALMA [C ii] line and far-infrared (FIR) continuum observations of three $z \gt 6$ low-luminosity quasars ($M_{\rm 1450} \gt -25$ mag) discovered by our Subaru Hyper Suprime-Cam (HSC) survey. The [C ii] line was detected in all three targets with luminosities of $(2.4\mbox{--}9.5) \times 10^8\, L_{\odot }$, about one order of magnitude smaller than optically luminous ($M_{\rm 1450} \lesssim -25$ mag) quasars. The FIR continuum luminosities range from $\lt 9 \times 10^{10}\, L_{\odot }$ (3 $\sigma$ limit) to ${\sim } 2 \times 10^{12}\, L_{\odot }$, indicating a wide range in star formation rates in these galaxies. Most of the HSC quasars studied thus far show [C ii]/ FIR luminosity ratios similar to local star-forming galaxies. Using the [C ii]-based dynamical mass ($M_{\rm dyn}$) as a surrogate for bulge stellar mass ($M_{\rm\, bulge}$), we find that a significant fraction of low-luminosity quasars are located on or even below the local $M_{\rm\, BH}$–$M_{\rm\, bulge}$ relation, particularly at the massive end of the galaxy mass distribution. In contrast, previous studies of optically luminous quasars have found that black holes are overmassive relative to the local relation. Given the low luminosities of our targets, we are exploring the nature of the early co-evolution of supermassive black holes and their hosts in a less biased way. Almost all of the quasars presented in this work are growing their black hole mass at a much higher pace at $z \sim 6$ than the parallel growth model, in which supermassive black holes and their hosts grow simultaneously to match the local $M_{\rm\, BH}$–$M_{\rm\, bulge}$ relation at all redshifts. As the low-luminosity quasars appear to realize the local co-evolutionary relation even at $z \sim 6$, they should have experienced vigorous starbursts prior to the currently observed quasar phase to catch up with the relation.