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).
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).