The Thermodynamic Cost of Computing with Heat
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 $\langle ξ\rangle$ diverges as $\langle ξ\rangle$ 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. We demonstrate that the required dissipation grows linearly $\mathcal{O}(L)$ with network depth for shallow networks, but diverges as the network depth $L$ approaches a fundamental maximum limit $L_{\max} = \epsNet/\epsMin$ imposed by the minimum error floor. Furthermore, under strong noise amplification conditions, the required dissipation can grow up to $\mathcal{O}(L^3)$. Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.