Optimal Computation from Fluctuation Responses

TL;DR

This work tackles the energy cost of computation in nonequilibrium systems by introducing a unified FRR-based framework that jointly optimizes the control protocol and the resulting probability distribution via trajectory data. By encoding information through coarse-graining and designing task-specific loss functions, the method uses FRR-derived gradients to drive systems toward target informational states while minimizing work, applicable to both overdamped and underdamped dynamics. The authors demonstrate the approach on canonical tasks, including bit erasure in a double-well potential and translating harmonic traps, and extend it to underdamped dynamics for bit-flip operations, achieving work costs near the Landauer bound $k_B T \,\log 2$. Overall, the framework provides principled, gradient-based strategies for thermodynamically efficient information processing with broad potential applications in quantum, chemical, and biological contexts.

Abstract

The energy cost of computation has emerged as a central challenge at the intersection of physics and computer science. Recent advances in statistical physics -- particularly in stochastic thermodynamics -- enable precise characterizations of work, heat, and entropy production in information-processing systems driven far from equilibrium by time-dependent control protocols. A key open question is then how to design protocols that minimize thermodynamic cost while ensur- ing correct outcomes. To this end, we develop a unified framework to identify optimal protocols using fluctuation response relations (FRR) and machine learning. Unlike previous approaches that optimize either distributions or protocols separately, our method unifies both using FRR-derived gradients. Moreover, our method is based primarily on iteratively learning from sampled noisy trajectories, which is generally much easier than solving for the optimal protocol directly from a set of governing equations. We apply the framework to canonical examples -- bit erasure in a double-well potential and translating harmonic traps -- demonstrating how to construct loss functions that trade-off energy cost against task error. The framework extends trivially to underdamped systems, and we show this by optimizing a bit-flip in an underdamped system. In all computations we test, the framework achieves the theoretically optimal protocol or achieves work costs comparable to relevant finite time bounds. In short, the results provide principled strategies for designing thermodynamically efficient protocols in physical information-processing systems. Applications range from quantum gates robust under noise to energy-efficient control of chemical and synthetic biological networks.

Figures (6)

• Figure 1: Optimal protocol for translating a harmonic trap: We fix $a_f=1$ and $\tau=1$ and use $10$ break points. For one training step, we simulate $5000$ trajectories with protocol value $\boldsymbol{a}=\{a_{1}, \dots, a_{10}\}$ to compute the gradient of the loss functions and use the gradient to update $\boldsymbol{a}$ and we train the protocol for $600$ iterations. We numerically show different types of jump with different weight combinations.
• Figure 2: Bit erasure optimization: For each training iteration, we simulate $5000$ trajectories to compute the gradient and update parameters. We validate with $10^5$ trajectories to estimate work cost and error. The number of error trajectories almost vanishes after $100$ iterations. Three trajectory ensembles are selected from a random training. The plot also includes the upper and lower bound of the minimal work cost in finite-time bit erasure under full control of the potential proesmans2020finite, denoted with a gray highlight.
• Figure 3: Error rate and work dependence on the hyperparameter ratio in erasure: Around $\alpha_{w}/\alpha_{1}= 0.47$, we observe a sharp rise in error rate. The inset is error rate and work versus $(\alpha_{w}/\alpha_{1}- 0.47)$ in log scale, indicating a second order phase transition.
• Figure 4: Optimal erasure protocol: For each protocol parameter {$a_{t}$, $b_t$, $c_t$}, we use $10$ break points for parameterization. (a) They are initialized to $0$. (b) Final protocol values after training for $1000$ iterations. (c) Potentials $U$ at different times.
• Figure 5: (a) Bit-flip mean work: Initialization of the potential is $U(x,t)=0$ for $t\in(0,1)$. Gray dashed line is the work cost of protocol proposed in Ref. ray2021non, which is $0.725 k_{\text{B}}T$. Trajectories are chosen from a single random training. (b) Phase space trajectories: Two trajectory classes in the whole phase space---one cluster starts near $-1$ and the other starts near $+1$.