  We report our first discoveries of high-redshift supernovae from the Subaru
HIgh-Z sUpernova CAmpaign (SHIZUCA), the transient survey using Subaru/Hyper
Suprime-Cam. We report the discovery of three supernovae at the
spectroscopically-confirmed redshifts of 2.399 (HSC16adga), 1.965 (HSC17auzg),
and 1.851 (HSC17dbpf), and two supernova candidates with the host-galaxy
photometric redshifts of 3.2 (HSC16apuo) and 4.2 (HSC17dsid), respectively. In
this paper, we present their photometric properties and the spectroscopic
properties of the confirmed high-redshift supernovae are presented in the
accompanying paper (Curtin et al. 2019). The supernovae with the confirmed
redshifts of z ~ 2 have the rest ultraviolet peak magnitudes close to -21 mag
and they are likely superluminous supernovae. The discovery of three supernovae
at z ~ 2 roughly corresponds to the approximate event rate of ~ 900 +/- 520
Gpc-3 yr-1 with Poisson error, which is consistent with the total superluminous
supernova rate estimated by extrapolating the local rate based on the cosmic
star-formation history. Adding unconfirmed superluminous supernova candidates
would increase the event rate. Our superluminous supernova candidates at the
redshifts of around 3 and 4 indicate the approximate superluminous supernova
rates of ~ 400 +/- 400 Gpc-3 yr-1 (z ~ 3) and ~ 500 +/- 500 Gpc-3 yr-1 (z ~ 4)
with Poisson errors. Our initial results demonstrate the outstanding capability
of Hyper Suprime-Cam to discover high-redshift supernovae.
