Is “Shannon-capacity of noisy computing” zero?

Towards understanding energy requirements for computation with noisy gates, we consider the computation of an arbitrary k-input, k-output binary invertible function. For broader applicability of our results, we allow using some noiseless gates in conjugation with noisy ones. However, we stipulate that the input gates on the computation graph must be separated from the output nodes by a “noisy cut.” We show that for the “information-friction” model proposed recently for energy consumed in circuits, and for binary-input AWGN noise in the computational nodes (with fixed communication schedule), the total (gate + info-friction) energy consumption for reliable computation must diverge to infinity as the target error probability is lowered to zero. Thus, in this model, the capacity of noisy computing is zero regardless of how “rate” of computation is defined: the required energy is unbounded for reliable computation regardless of how efficiently the available resources (e.g. gates, wires, etc.) are used.