That's not how this stuff works. You would use some sort of large capacitor/battery bank which is charged by a much smaller power draw over a long time, and then dump the energy over a short time.
The laser array currently operating at the National Ignition Facility has a peak output of 500 TW, more than three orders of magnitude higher than the number you called ludicrous. Obviously Starshot would be applying that power for a much longer time interval, and indeed would be a global undertaking dwarfing NIF. But it doesn't need 100GW of dedicated power plants.