  Periodically variable quasars have been suggested as close binary
supermassive black holes. We present a systematic search for periodic light
curves in 625 spectroscopically confirmed quasars with a median redshift of 1.8
in a 4.6 deg$^2$ overlapping region of the Dark Energy Survey Supernova
(DES-SN) fields and the Sloan Digital Sky Survey Stripe 82 (SDSS-S82). Our
sample has a unique 20-year long multi-color ($griz$) light curve enabled by
combining DES-SN Y6 observations with archival SDSS-S82 data. The deep imaging
allows us to search for periodic light curves in less luminous quasars (down to
$r{\sim}$23.5 mag) powered by less massive black holes (with masses
$\gtrsim10^{8.5}M_{\odot}$) at high redshift for the first time. We find five
candidates with significant (at $>$99.74% single-frequency significance in at
least two bands with a global p-value of
$\sim$7$\times10^{-4}$--3$\times10^{-3}$ accounting for the look-elsewhere
effect) periodicity with observed periods of $\sim$3--5 years (i.e., 1--2 years
in rest frame) having $\sim$4--6 cycles spanned by the observations. If all
five candidates are periodically variable quasars, this translates into a
detection rate of ${\sim}0.8^{+0.5}_{-0.3}$% or ${\sim}1.1^{+0.7}_{-0.5}$
quasar per deg$^2$. Our detection rate is 4--80 times larger than those found
by previous searches using shallower surveys over larger areas. This
discrepancy is likely caused by differences in the quasar populations probed
and the survey data qualities. We discuss implications on the future direct
detection of low-frequency gravitational waves. Continued photometric
monitoring will further assess the robustness and characteristics of these
candidate periodic quasars to determine their physical origins.
