Balancing Error and Dissipation in Highly-Reliable Computing

Modern digital electronics support remarkably reliable computing, especially given the challenge of controlling nanoscale logical components that interact in fluctuating environments. However, the high-reliability limit is subject to a fundamental error-energy-efficiency tradeoff that arises from time-symmetric control. Requiring a low probability of error causes energy consumption to diverge as logarithm of the inverse error rate for nonreciprocal logical transitions. The reciprocity (self-invertibility) of a computation is a stricter condition for thermodynamic efficiency than logical reversibility (invertibility), the latter being the root of Landauer's work bound. In fact, the average energy required for reliable erasure is well above that bound. Explicit simulations of work production closely track the error-dissipation tradeoff, diverging from the Landauer bound as the error rate decreases. And, unlike the Landauer work, which can be recovered, the nonreciprocal work must be irreversibly dissipated. Analogous bounds hold for the universal NAND gate and extend to circuits of logical gates. That said, strictly-reciprocal logic gates, such as communication relays and NOT gates, are exempt and can be efficiently implemented via time-symmetric protocols. For all other computations, though, time-asymmetric control must be used to avoid dissipation arising from nonreciprocity. The lesson is that highly-reliable computation under time-symmetric control cannot reach, and is often far above, the Landauer limit. In this way, time-asymmetry becomes a design principle for thermodynamically-efficient computing. We also demonstrate that the time-reversal symmetries of the memory elements themselves play an essential role in determining the minimal energy necessary to implement a computation.