The Thermodynamic Cost of Computing with Heat

Abstract

Autonomous quantum thermal machines have recently been proposed as physics-based computing substrates where logical inputs and outputs are encoded in temperature gradients. While such ``thermodynamic neurons'' exhibit a clear trade-off between computational fidelity and heat dissipation, the fundamental information-theoretic limits of temperature-encoded computation remain uncharacterized. Here, we derive rigorous bounds linking average error probability, channel capacity, and entropy production for finite-capacity thermal reservoirs operating far from equilibrium. We prove that the minimal dissipation required to achieve a target average error probability $\langle \xi \rangle$ diverges as $\langle \xi \rangle$ approaches a fundamental minimum error floor $\epsMin$ imposed by finite-reservoir thermal fluctuations. We further establish a thermodynamic channel capacity that saturates at high dissipation, and quantify the minimal dissipation required for cascaded networks to maintain target fidelity, demonstrating a fundamental $\mathcal{O}(L \ln L)$ overhead with network depth, with the required dissipation growing up to $\mathcal{O}(L^3)$ under strong noise amplification conditions. Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.