# Exact, complete expressions for the thermodynamic costs of circuits

Common engineered systems implement computations using circuits, as do many biological systems (e.g., gene regulatory networks). The topology of such circuits introduces constraints on the physical system implementing the circuit. Here we derive the addition to the minimal heat flow out of a system that implements a desired computation using a given circuit rather than using an idealized system that has no constraints. We call this additional minimal heat flow "circuit Landauer loss". We show that in general different circuits computing the same function have different circuit Landauer loss. This leads to a novel optimization problem, of how to design a circuit to compute a given function with minimal circuit Landauer loss. We also analyze a second contribution to the actual heat that flows out of a circuit when running, beyond the minimum possible. We call this "circuit mismatch loss". It is the extra heat that is dissipated when a physical circuit that is designed to dissipate least heat when it is run with a distribution q over its inputs, is instead run with a distribution p q over its inputs. Circuit mismatch loss can be either positive or negative. Indeed, we show that the sum of circuit Landauer loss and circuit mismatch loss can be negative. So in practice, using a circuit to compute a function, rather than a fixed "idealized system" that has no constraints, can actually reduce the total heat produced.

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