The Energy-Advantage Frontier of Random Circuit Sampling Across the Simulability Transition

We introduce the energy-advantage frontier of random circuit sampling (RCS)—the locus in the noise-rate/size plane where the classical and quantum energy-to-solution are equal—and locate it relative to the noise-induced classical-simulability transition. Our firm, model-free result is that the classical sampling-energy floor collapses sharply across the transition, exhibiting a finite-size scaling collapse over exact systems n≤12 (with a collapse exponent of order unity). Built on it, the energy-advantage frontier is distinct from the computational (wall-clock) frontier: in the exascale-classical, few-femtojoule regime, the energy frontier is the more noise-robust, so an energy advantage can persist where a time advantage does not—a projection, not a deployment claim, whose direction reverses outside that regime. We give a conditional Landauer lower bound for sampling, RE≡Ecl/Eq≳2(1.0±0.2)n/n under standard RCS hardness, package RE as a single figure of merit computed by a released open tool (exs), and map an energy-normalized cross-entropy witness within a cost model. A run on a 156-qubit IBM Heron processor tests the fidelity model (no energy is measured on hardware) and shows real noise exceeds the depolarizing idealization, so quoted frontier locations are upper bounds.