Because thermodynamics and Shannon entropy are equivalent, all computationally reversible processes are thermodynamically reversible as well, at least in principle. Thus, you only need to "consume" power when doing a destructive update (i.e., overwriting memory locations) - and the minimum amount of energy necessary to do this per-bit is known, just like the maximum efficiency of a heat engine is known.

Of course, for a closed timelike loop, the entire process has to return to its start state, which means there is theoretically zero net energy loss (otherwise the loop wouldn't be stable).