Machine learning is typically described in terms of deterministic logical operations, whereas physical systems generally operate in the presence of noise, dissipation and irreversibility. Here, we turn these physical effects into computational resources for an autonomous, interpretable learning architecture. We develop a classifier based on thermodynamic neurons, which are autonomous quantum thermal machines that implement logical operations through heat flow, and use these to construct a stochastic version of the Tsetlin machine, an interpretable rule-based learning architecture. By combining thermodynamic AND, NOT and OR gates with an autonomous coupling mechanism, we realise a learning system whose computation unfolds without the need for external time-dependent control. Despite its noisy components, the resulting classifier achieves classification accuracy that is statistically comparable to that of the standard Tsetlin machine. Reliability arises from architectural mechanisms such as thresholding and redundancy, rather than exact logical operations. Our results highlight that accurate and interpretable learning can emerge from autonomous stochastic dynamics, and establish thermodynamic computation as a viable framework for physical machine learning.
Comments
12 pages + 3 pages of Appendices + 3 pages of Suppl. Materials; 5 figures + 2 in the Suppl. Materials
Machine learning is typically described in terms of deterministic logical operations, whereas physical systems generally operate in the presence of noise, dissipation and irreversibility. Here, we turn these physical effects into computational resources for an autonomous, interpretable learning architecture. We develop a classifier based on thermodynamic neurons, which are autonomous quantum thermal machines that implement logical operations through heat flow, and use these to construct a stochastic version of the Tsetlin machine, an interpretable rule-based learning architecture. By combining thermodynamic AND, NOT and OR gates with an autonomous coupling mechanism, we realise a learning system whose computation unfolds without the need for external time-dependent control. Despite its noisy components, the resulting classifier achieves classification accuracy that is statistically comparable to that of the standard Tsetlin machine. Reliability arises from architectural mechanisms such as thresholding and redundancy, rather than exact logical operations. Our results highlight that accurate and interpretable learning can emerge from autonomous stochastic dynamics, and establish thermodynamic computation as a viable framework for physical machine learning.
We develop a physics-based model for classical computation based on autonomous quantum thermal machines. These machines consist of few interacting quantum bits (qubits) connected to several environments at different temperatures. Heat flows through the machine are here exploited for computing. The process starts by setting the temperatures of the environments according to the logical input. The machine evolves, eventually reaching a nonequilibrium steady state, from which the output of the computation can be determined via the temperature of an auxilliary finite-size reservoir. Such a machine, which we term a "thermodynamic neuron," can implement any linearly separable function, and we discuss explicitly the cases of NOT, 3-MAJORITY, and NOR gates. In turn, we show that a network of thermodynamic neurons can perform any desired function. We discuss the close connection between our model and artificial neurons (perceptrons) and argue that our model provides an alternative physics-based analog implementation of neural networks, and more generally a platform for thermodynamic computing.
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 ⟨ξ⟩ diverges as ⟨ξ⟩ 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 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 O(L3). Our framework bridges stochastic thermodynamics, finite-time information theory, and autonomous computation, providing rigorous design principles for energy-efficient analog thermodynamic hardware.
We develop a physics-based model for classical computation based on autonomous quantum thermal machines. These machines consist of few interacting quantum bits (qubits) connected to several environments at different temperatures. Heat flows through the machine are here exploited for computing. The process starts by setting the temperatures of the environments according to the logical input. The machine evolves, eventually reaching a non-equilibrium steady state, from which the output of the computation can be determined via the temperature of an auxilliary finite-size reservoir. Such a machine, which we term a "thermodynamic neuron", can implement any linearly-separable function, and we discuss explicitly the cases of NOT, 3-majority and NOR gates. In turn, we show that a network of thermodynamic neurons can perform any desired function. We discuss the close connection between our model and artificial neurons (perceptrons), and argue that our model provides an alternative physics-based analogue implementation of neural networks, and more generally a platform for thermodynamic computing.
Thermodynamics is a theory of impressive success and a wide range of applicability. Nevertheless, it took about two hundred years after the basic formulation of phenomenological thermodynamics until Boltzmann and Maxwell gave a foundation in terms statistical mechanics based on classical mechanics. The invention of quantum mechanics has triggered various attempts to establish a theory of quantum thermodynamics, i.e., to explain and derive thermodynamics by quantum mechanics only. Recently, a new approach to this question has been developed by Gemmer et al. focusing on the role of the partitioning of the universe into "system" and "environment" and on how the entanglement between those parts and the properties of typical environments lead to a thermal state in the system for almost any instant in time. Within this approach, the emergence of thermodynamic behavior in subsystems of a wide class of autonomous quantum networks could be established although the state of the total system is a pure. This turns our understanding of the emergence of thermodynamics upside down: No longer the presence of ideal (infinitely large and always in equilibrium) makes the system thermodynamic, but thermodynamic behavior of the system itself induced by an appropriate embedding - which by itself does not need to be thermodynamic in any way - becomes the crucial ingredient. Moreover, it was found that not only macroscopic embeddings but also quantum networks as small as 10 spins may serve already as excellent thermal embeddings. In this context, this thesis approaches a number of questions: First, complementary to the emergence of relaxation (heat), how does mechanical control over a system (work) emerge in autonomous quantum systems? What types of embeddings allow for this control, and is there a lower limit of their size? The answer is discussed by making the connection between work, parametric control of a Hamiltonian, and classical driving with the help of the factorization approximation. Using this link, we propose the local effective measurement basis method to determine heat and work currents in arbitrary bipartite quantum systems, and discuss measures of how to assess the thermodynamic functionality of an arbitrary embedding. We then apply these concepts to a minimal model consisting of only a spin and a quantum harmonic oscillator (spin-oscillator model) to demonstrate that even a single oscillator may act as an ideal work reservoir. A slight variation of the model then illustrates the pros and cons of the proposed measures of thermodynamic functionality. Second, since thermodynamics is first and foremost a theory of processes, which need both, thermal and mechanical control, we ask if autonomous quantum networks can implement thermodynamic cycles. We do so by first reviewing some specialties of quantum thermodynamic processes in general and then presenting the autonomous dynamic three spin machine, an autonomous quantum network implementing a thermodynamic cycle, which is driven by a emergent quantum work reservoir as discussed in the first part. Finally, we discuss quantum thermodynamic pseudomachines, a class of models that exhibit machine-like functionality without use of a thermodynamic cycle. This gives an answer to the question how new forms of control that are only found in quantum systems lead to new thermodynamic functionalities. The presented models can all be understood as thermodynamic laser models. In particular, we discuss the extended dissipative Jaynes-Cummings model and its thermodynamic properties. We resolve a conflict between different interpretations of the model by a careful analysis of the model itself and the thermodynamic concepts used by different authors. It is found that the model is an intricate heat transport model and no machine and that its thermodynamic functionality relies on the transition-selective coupling of heat baths to a few-level quantum system.
A Boltzmann machine is a stochastic neural network that has been extensively used in the layers of deep architectures for modern machine learning applications. In this paper, we develop a Boltzmann machine that is capable of modeling thermodynamic observables for physical systems in thermal equilibrium. Through unsupervised learning, we train the Boltzmann machine on data sets constructed with spin configurations importance sampled from the partition function of an Ising Hamiltonian at different temperatures using Monte Carlo (MC) methods. The trained Boltzmann machine is then used to generate spin states, for which we compare thermodynamic observables to those computed by direct MC sampling. We demonstrate that the Boltzmann machine can faithfully reproduce the observables of the physical system. Further, we observe that the number of neurons required to obtain accurate results increases as the system is brought close to criticality.
Quantum coherence has been shown to impact the operational capabilities of quantum systems performing thermodynamic tasks in a significant way, and yet the possibility and conditions for genuine coherence-enhanced thermodynamic operation remain unclear. Introducing a comparison with classical machines using the same set of thermodynamic resources, we show that for steady-state quantum thermal machines – both autonomous and externally driven – that interact weakly with thermal reservoirs and work sources, the presence of coherence induced by perturbations in the machine Hamiltonian guarantees a genuine thermodynamic advantage under mild conditions. This advantage applies to both cases where the induced coherence is between levels with different energies or between degenerate levels. On the other hand, we show that engines subjected to noise-induced coherence can be outperformed by classical stochastic engines using exactly the same set of (incoherent) resources. We illustrate our results with three prototypical models of heat engines and refrigerators: the three-level amplifier, the three-qubit autonomous refrigerator, and a noise-induced-coherence machine.