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.
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