The Fluctuation-Dissipation Compilation Theorem: A Unified Framework for Thermodynamically Optimal Computing

Computation fundamentally requires energy, with Landauer's principle establishing that erasing one bit of information irreversibly dissipates at least thermal energy times the natural logarithm of two. Modern computers operate millions to billions of times above this fundamental limit. Here we prove the Fluctuation-Dissipation Compilation Theorem, establishing rigorous bounds for transforming arbitrary irreversible programs into thermodynamically reversible equivalents that approach the Landauer limit with quantifiable precision. Our central result unifies fluctuation theorems from non-equilibrium statistical mechanics with compilation theory from computer science. We prove that for any computation producing logical entropy change—measured as Shannon entropy of computational basis measurement distributions—there exists a stochastic reversible compilation whose average energy dissipation equals the product of temperature and logical entropy change, plus corrections that decrease as the inverse square root of the ensemble size. This bound is provably tight: it is achievable through Bennett's copy-compute-uncompute strategy and cannot be improved asymptotically by any alternative method. The proof combines the Jarzynski equality and Crooks fluctuation relation with the central limit theorem and Berry-Esseen convergence bounds, establishing explicit convergence rates absent from prior reversible computing theory. We formulate thermodynamic compilation as a resource theory where reversible transformations conserve entropy and irreversible erasure operations incur fundamental work costs bounded by Shannon information limits. Beyond Bennett's asymptotic results, we provide finite-ensemble guarantees essential for practical implementations, characterizing work fluctuations through complete statistical distributions rather than average values alone. We establish a comprehensive theoretical hierarchy encompassing ancilla requirements by function class and work-time-space Pareto frontiers proving no compilation can simultaneously optimize all resources. Bennett's method offers two implementation variants: standard compilation achieves linear time overhead with space proportional to computation length, while recursive pebbling reduces space to logarithmic growth at quadratic time cost. For fault-tolerant quantum computing, we prove quantum error correction overhead scales with circuit dimensions rather than logical entropy—deterministic algorithms with zero entropy change still incur syndrome measurement costs proportional to circuit size, code distance squared, and physical error rate. Classical simulation of idealized quantum circuits under perfect reversibility assumptions validates our theoretical predictions: computed dissipation for deterministic algorithms approaches effectively zero beyond numerical precision limits, while probabilistic algorithms exhibit dissipation proportional to their logical entropy production. This framework provides the first complete, quantitatively proven path from fundamental physical limits to practical energy-optimal compilation. Applications span quantum computing architectures where thermal budgets constrain gate fidelities, energy-harvesting Internet-of-Things devices requiring decade-scale battery operation, data centers seeking efficiency improvements with billion-dollar economic impact, and deep-space missions operating under extreme power constraints. By establishing thermodynamic compilation as a rigorous mathematical discipline, this work bridges statistical physics, information theory, and computer science, opening new research directions in the ultimate physical limits of computation.