The unique combination of superb spatial resolution, wide-area coverage, and deep depth of the optical imaging from the Hyper Suprime-Cam (HSC) Subaru Strategic Program is utilized to search for dual quasar candidates. Using an automated image analysis routine on 34,476 known Sloan Digital Sky Survey quasars, we identify those with two (or more) distinct optical point sources in HSC images covering 796 deg(2). We find 421 candidates out to a redshift of 4.5 of which one hundred or so are more likely after filtering out contaminating stars. Angular separations of 06-40 correspond to projected separations of 3-30 kpc, a range relatively unexplored for population studies of luminous dual quasars. Using Keck I/Low Resolution Imaging Spectrometer and Gemini-N/Near-Infrared Integral Field Spectrometer, we spectroscopically confirm three dual quasar systems atz < 1, two of which are previously unknown out of eight observed, based on the presence of characteristic broad emission lines in each component, while highlighting that the continuum of one object in one of the pairs is reddened. In all cases, the [Oiii]lambda 5007 emission lines have mild velocity offsets, thus the joint [Oiii] line profile is not double-peaked. We find a dual quasar fraction of 0.26 0.18% and no evidence for evolution. A comparison with the Horizon-AGN simulation seems to support the case of no evolution in the dual quasar fraction when broadly matching the quasar selection. These results may indicate a scenario in which the frequency of the simultaneous triggering of luminous quasars is not as sensitive as expected to the cosmic evolution of the merger rate or gas content of galaxies.