  We present ALMA [CII] line and far-infrared (FIR) continuum observations of
three $z > 6$ low-luminosity quasars ($M_{\rm 1450} > -25$) discovered by our
Subaru Hyper Suprime-Cam (HSC) survey. The [CII] line was detected in all three
targets with luminosities of $(2.4 - 9.5) \times 10^8~L_\odot$, about one order
of magnitude smaller than optically luminous ($M_{\rm 1450} \lesssim -25$)
quasars. The FIR continuum luminosities range from $< 9 \times 10^{10}~L_\odot$
(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 [CII]/FIR luminosity ratios similar to local star-forming galaxies.
Using the [CII]-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 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.
