  Observations in the local universe show a tight correlation between the
masses of supermassive black holes (SMBHs; $M_{\rm BH}$) and host-galaxy bulges
($M_{\rm bulge}$), suggesting a strong connection between SMBH and bulge
growth. However, direct evidence for such a connection in the distant universe
remains elusive. We have studied sample-averaged SMBH accretion rate
($\overline{\rm BHAR}$) for bulge-dominated galaxies at $z=0.5-3$. While
previous observations found $\overline{\rm BHAR}$ is strongly related to
host-galaxy stellar mass ($M_\star$) for the overall galaxy population, our
analyses show that, for the bulge-dominated population, $\overline{\rm BHAR}$
is mainly related to SFR rather than $M_\star$. This ${\overline{\rm
BHAR}}$-SFR relation is highly significant, e.g. $9.0\sigma$ (Pearson
statistic) at $z=0.5-1.5$. Such a $\overline{\rm BHAR}$-SFR connection does not
exist among our comparison sample of galaxies that are not bulge-dominated, for
which $M_\star$ appears to be the main determinant of SMBH accretion. This
difference between the bulge-dominated and comparison samples indicates that
SMBHs only coevolve with bulges rather than the entire galaxies, explaining the
tightness of the local $M_{\rm BH}$-$M_{\rm bulge}$ correlation. Our best-fit
$\overline{\rm BHAR}$-SFR relation for the bulge-dominated sample is
$\log\overline{\rm BHAR} = \log\mathrm{SFR} - (2.48\pm0.05)$ (solar units). The
best-fit $\overline{\rm BHAR}/\mathrm{SFR}$ ratio ($10^{-2.48}$) for
bulge-dominated galaxies is similar to the observed $M_{\rm BH}/M_{\rm bulge}$
values in the local universe. Our results reveal that SMBH and bulge growth are
in lockstep, and thus non-causal scenarios of merger averaging are unlikely the
origin of the $M_{\rm BH}$-$M_{\rm bulge}$ correlation. This lockstep growth
also predicts that the $M_{\rm BH}$-$M_{\rm bulge}$ relation should not have
strong redshift dependence.
