Balancing Error and Dissipation in Computing

Abstract

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. Beyond engineered computation, the results reveal a generic error-dissipation tradeoff in steady-state transformations of genetic information as carried out by biological organisms.

Initial-state dependence of thermodynamic dissipation for any quantum process

“…This is consistent with the generic error-dissipation tradeoff recently discovered for non-reciprocated computations in Ref. [44], but only explains the error-dissipation tradeoff for logically irreversible transitions like erasure. For logically reversible but nonreciprocal transitions, all initial distributions suffer the same error-dissipation tradeoff.…”

“…In those cases, the error-dissipation tradeoff is not a consequence of the contraction of the relative entropy discussed here, but rather follows more generally from the theory laid out in Ref. [44].…”

“…Under control constraints-like time-symmetric drivingwhere fidelity costs significant dissipation [44], we find that q 0 may be forced to have eigenvalues of order and thus D ρ 0 q 0 can diverge as ln(1/ ). This is consistent with the generic error-dissipation tradeoff recently discovered for non-reciprocated computations in Ref.…”

Non-Markovian Momentum Computing: Universal and Efficient

“…This suggests the existence of a lower bound on entropy production-one that accounts for the coursegraining, as predicted in Ref. [28]. Conclusion Rate equation dynamics is certainly a venerable and powerful framework, central to reaction kinetics in chemistry [29,30] and key to the master equations of applied statistical mechanics [2,14,15].…”

Finite-time Landauer principle

“…Within the fields of finite-time thermodynamics [13] and stochastic thermodynamics [14,15] the search for protocols that minimize the average dissipation of a mesoscopic thermodynamic system during finite-time transformations has focused on the optimization of a finite (and usually small) number of control parameters influencing the potential landscape of the system [16][17][18][19][20][21]. For bit erasure, limiting control to a fixed set of parameters may make it more costly or even impossible to fully erase a bit [22][23][24]. * email: Karel Proesmans@sfu.ca An important advance is the work of Aurell et al [25,26], which uses full control over the potential landscape to find protocols valid in both slow and fast limits that minimize entropy production for a final state constrained to a fixed microscopic probability distribution.…”