Approaching the Landauer Limit: Thermodynamically Optimal Compilation with Explicit Convergence Rates

Computation requires energy. Landauer’s principle establishes that erasing one bit of information irreversibly dissipates at least kBT ln 2 of thermal energy. Modern computers operate 10^6 to 10^9 times above this fundamental bound. In 1973, Bennett demonstrated that any computation can be made thermodynamically reversible through procedures that preserve information, thereby approaching the Landauer limit. However, a critical question remained unanswered for fifty years: how quickly does reversible compilation converge to the theoretical minimum, and what ensemble size achieves practical energy savings? We resolve this problem through the Fluctuation-Dissipation Compilation Theorem, which provides the first explicit convergence rate for thermodynamically optimal computing. Reversible compilation of any algorithm dissipates average energy equal to the thermal energy scale multiplied by the logical entropy change, plus corrections that decrease as the inverse square root of the ensemble size. This N^(−1/2) convergence rate, derived by combining fluctuation theorems from statistical mechanics with the central limit theorem, is provably optimal—no compilation strategy can achieve faster convergence. The proof unifies three historically independent disciplines: statistical mechanics through the Jarzynski equality and Crooks fluctuation theorem, information theory through Shannon entropy bounds, and compilation theory through Bennett’s reversible computing framework. By formulating thermodynamic compilation as a rigorous resource theory, we establish that reversible transformations exactly conserve entropy while irreversible erasure operations incur fundamental energy costs bounded by information-theoretic limits. Critically, we provide finite-ensemble performance guarantees essential for practical implementations, characterizing not just average energy dissipation but complete statistical distributions including fluctuations and tail probabilities. Numerical experiments on Grover search and quantum Fourier transforms validate our predictions. Simulations confirm the predicted N^(−1/2) convergence rate within 3% error and demonstrate energy reductions exceeding 99.9% compared to irreversible implementations. These results establish that thermodynamically optimal compilation is achievable with realistic ensemble sizes for near-term quantum hardware operating under thermal constraints. This framework bridges fundamental physics and practical computing. Applications span quantum processors cooled to millikelvin temperatures where every picojoule of dissipation limits performance, battery-powered Internet-of-Things sensors requiring decade-long autonomous operation, data centers consuming one percent of global electricity where incremental efficiency improvements yield billion-dollar savings, and deep-space missions operating on hundred-watt radioisotope generators where every watt conserved enables additional science. By establishing thermodynamic compilation as a rigorous engineering discipline with explicit finite-size guarantees, this work transforms Bennett’s asymptotic theoretical insight into a practical methodology for approaching fundamental physical limits in real computing systems