Detectability of 21cm-signal during the Epoch of Reionization with
  21cm-Lyman-$\alpha$ emitter cross-correlation. I
  Detection of the redshifted 21cm-line signal from neutral hydrogen in the
intergalactic medium (IGM) during the Epoch of Reionization (EoR) is
complicated by intense foregrounds such as galactic synchrotron and
extragalactic radio galaxies. The 21cm-Lyman-$\alpha$ emitter(LAE)
cross-correlation is one of the tools available to reduce the foreground
effects because the foreground emission from such radio sources is
statistically independent of LAE distribution. LAE surveys during the EoR at
redshifts $z=6.6$ and $7.3$ are ongoing by the Subaru Hyper Suprime-Cam (HSC).
Additionally, Prime Focus Spectrograph (PFS) will provide precise redshift
information of the LAEs discovered by the HSC survey. In this paper, we
investigate the detectability of the 21cm signal with the 21cm-LAE
cross-correlation by using our improved reionization simulations. We also focus
on the error budget and evaluate it quantitatively in order to consider a
strategy to improve the signal-to-noise ratio. In addition, we explore an
expansion of the LAE survey to suggest optimal survey parameters and show a
potential to measure a characteristic size of ionized bubbles via the turnover
scale of the cross-power spectrum. As a result, we find that the Murchison
Widefield Array (MWA) has ability to detect the cross-power spectrum signal on
large scales by combining LAE Deep field survey of HSC. We also show that the
sensitivity is improved dramatically at small scales by adding redshift
information from the PFS measurements. The Square Kilometre Array (SKA) has a
potential to measure the turnover scale with an accuracy of
$6\times10^{-3}~{\rm Mpc^{-1}}$.
