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Nanostructure and nanoantenna-based all-optical (AO) devices have attracted significant research interests in recent years due to their small size, high information capacity, ultrafast processing, low power consumption, and overall practicality. Here, in this Letter, we propose a novel metasurface having quasi-rhombus-shaped antennas to modulate optical modes in a dielectric-loaded waveguide for the realization of a complete family of logic gates including NOT, AND, OR, XOR, NAND, NOR, and XNOR. These logic operations are realized using destructive and constructive interferences between the input optical signals. The high contrast ratios of about 33.39, 27.69, and 33.11 dB are achieved for the NAND, NOR, and XNOR logic gates, respectively, with the speed as high as 108 Gb/s.