Fundamental limits on the thermodynamics of circuits

The thermodynamics of computation specifies the minimum amount that entropy must increase in the environment of any physical system that implements a given computation, when there are no constraints on how the system operates (the so-called “Landauer limit”). However common engineered computers use digital circuits that physically connect separate gates in a specific topology. Each gate in the circuit performs its own “local” computation, with no a priori constraints on how it operates. In contrast, the circuit’s topology introduces constraints on how the aggregate physical system implementing the overall “global” computation can operate. These constraints cause additional minimal entropy increase in the environment of the overall circuit, beyond that caused by the individual gates.

Here we analyze the relationship between a circuit’s topology and this additional entropy increase, which we call the “circuit Landauer cost”. We also compute a second kind of circuit cost, the “circuit mismatch cost”. This is the extra entropy that is generated if a physical system is designed to achieve minimal entropy production for a particular distribution q over its inputs, but is instead used with an input distribution p that differs from q.

We show that whereas the circuit Landauer cost cannot be negative, circuits can have positive or negative mismatch cost. In fact the total circuit cost (given by summing the two types of cost) can be positive or negative. Thus, the total amount of entropy increase in the environment can be either larger or smaller when a particular computation is implemented with a circuit.

Furthermore, in general different circuits computing the same Boolean function have both different Landauer costs and different mismatch costs. This provides a new set of challenges, never before considered, for how to design a circuit to implement a given computation with minimal thermodynamic cost. As a first step in addressing these challenges, we use tools from the computer science field of circuit complexity to analyze the scaling of thermodynamic costs for different computational tasks.