Does it imply the computer would not give off heat? Or that a significant amount of the heat generated by a computer is due to information loss?
It does, and there is a very strong (but not universally accepted) case that they are the same thing: https://en.wikipedia.org/wiki/Entropy_in_thermodynamics_and_...
Great lay article about it: http://lesswrong.com/lw/o5/the_second_law_of_thermodynamics_...
(Side note: I've long thought you can derive Carnot efficiency based on information-theoretic arguments about the information contained in knowing only the temperature difference between a heat source and a sink.)
>Does it imply the computer would not give off heat? Or that a significant amount of the heat generated by a computer is due to information loss?
Basically, yes. You can't avoid the heat emission from setting a 0 to a 1 on that computer's storage medium, but you would avoid the loss from any of the intermediate computations that typically make CPUs so hot.
Wouldn't a reversible computer that executes need to take up an exponentially large amount of physical space as it executes?
That might work for small problems but it doesn't seem like it would scale. Or rather, it would scale to fill the known universe.
Sorry I haven't really dug into the attached articles yet, but I will.
The implication goes the other way: if the computer does not give off heat, it must be reversible.
An irreversible operation is anything that maps many (possible) states to the same state. Setting a bit to zero is irreversible (because it destroys the information about what bit was there previously). An AND gate is irreversible (because both TF and FT map to F). A NOT gate is reversible (because you can infer the input from the output.)
There are universal reversible gates (Toffoli and Fredkin gates are examples, but there are many others), so you can (theoretically) build a (usable) computer whose logical operations are completely reversible.