Evolutionary design of thermodynamic logic gates and their heat emission

TL;DR

The paper tackles the gap between Landauer's fundamental heat bound $k_{\rm B} T \ln 2$ and real-world dissipation by showing that control systems can be engineered to have heat outputs comparable to the information-carrying degrees of freedom. A mutation-based genetic algorithm trains a Langevin-based thermodynamic computer, composed of visible information units and hidden computational units, to perform erasure and XOR while controlling heat distribution. Results demonstrate that heat can be relocated away from the information register into the controller and, in some training regimes, even absorbed by visible units, enabling heat-management–aware computing architectures under stochastic thermodynamics. This approach points to practical pathways for energy-efficient thermodynamic computing where heat management is an intrinsic part of the program design.

Abstract

Landauer's principle bounds the heat generated by logical operations, but in practice the thermodynamic cost of computation is dominated by the control systems that implement logic. CMOS gates dissipate energy far above the Landauer bound, while laboratory demonstrations of near-Landauer erasure rely on external measurement or feedback systems whose energy costs exceed that of the logic operation by many orders of magnitude. Here we use simulations to show that a genetic algorithm can program a thermodynamic computer to implement logic operations in which the total heat emitted by the control system is of a similar order of magnitude to that of the information-bearing degrees of freedom. Moreover, the computer can be programmed so that heat is drawn away from the information-bearing degrees of freedom and dissipated within the control unit, suggesting the possibility of computing architectures in which heat management is an integral part of the program design.

Figures (4)

• Figure 1: (a) In this paper we consider simulations of a thermodynamic logic device built from visible and hidden units. Visible units feel a double-well potential described by the first sum in Eq. (\ref{['pot']}). These are information-storing units: logical states 0 and 1 correspond to the left- and right-hand portions of the double-well potential. Hidden units feel a (non-quadratic) single-well potential described by the third sum in Eq. (\ref{['pot']}); these are computational units. (b) We consider $N_{\rm v}$ visible units (blue) and $N_{\rm h}$ hidden units (black). There are connections $J_{ij}$ between visible and hidden units (blue), and between hidden units (black), described by the second sum in Eq. (\ref{['pot']}). These connections, and the hidden-unit biases $b_i$ described by the fourth sum in Eq. (\ref{['pot']}), are adjusted by genetic algorithm so that the logic device performs erasure (reset-to-zero) or XOR operations. For erasure and XOR we take $N_{\rm v}=1$ and 2, respectively, and in both cases we take $N_{\rm h}=10$.
• Figure 2: Program fidelity $P$ as a function of evolutionary time $n$ for (a) erasure and (b) XOR. Here the genetic algorithm is instructed to minimize the order parameter $\phi_1$, Eq. (\ref{['phi1']}).
• Figure 3: State-time probability densities $\rho(x_1,t)$ and $\rho(x_2,t)$ for the visible spins under the XOR program produced by minimizing $\phi_1$. The left (right) half of each box corresponds to logical state 0 (1); for the first visible spin, the dividing line is shown dotted. The inputs to the program are the logical states $S_1(0),S_2(0)$ of the two visible units at the initial time (top). The output of the program is the logical state $S_1(t_{\rm f})$ of the first visible unit at the final time (bottom). The four panels show the four input cases: (0,0) and (1,1) produce an output of 0, while (0,1) and (1,0) produce an output of 1. Probability densities are measured over $10^4$ independent trajectories.
• Figure 4: Histograms of total heat (black) and visible-unit heat (blue) emitted by the erasure program trained using the order parameters (a) $\phi_2$ and (b) $\phi_3$. Upon moving from one order parameter to the other, the total heat emission increases markedly, and the visible units go from emitting to absorbing heat. Histograms were taken over $10^4$ independent trajectories.