A typical integrated circuit pushes electrical voltage across pathways, with transistors and stuff modifying it. When you interpret those voltages as mathematical values and logical instructions, then congratulations, you have created a processor, memory, and so forth. You don't need to use electricity for this. In fact, the history of Charles Babbage and Ada Lovelace was their attempts to perform computation on mechanical state.
Image Credit: University of Colorado
Chip contains optical (left) and electric (top and right) circuits.
One possible follow-up is photonic integrated circuits. This routes light through optical waveguides, rather than typical electric traces. The prototype made by University of Colorado Boulder (and UC Berkeley) seem to use photonics just to communicate, and an electrical IC for the computation. The advantage is high bandwidth, high density, and low power.
This sort of technology was being investigated for several years. My undergraduate thesis for Physics involved computing light transfer through defects in a photonic crystal, using it to create 2D waveguides. With all the talk of silicon fabrication coming to its limits, as 14nm transistors are typically made of around two-dozen atoms, this could be a new direction to innovate.
And honestly, wouldn't you want to overclock your PC to 400+ THz? Make it go plaid for ludicrous speed. (Yes, this paragraph is a joke.)