What happens to a computer as it is cooled towards absolute zero?
Anything based on semiconductors would simply stop working, because they depend on electrons kicked into the conduction band by their own thermal energy.

Metallic conductors might become superconductors, but that doesn't mean anything if the first thing you hit is a chunk of inert silicon.

There are types of logic that are designed to work at cold temperatures and using superconducting materials, for example, Josephson junctions.