Thermodynamic Computing via Autonomous Quantum Thermal Machines

TL;DR

Thermodynamic computing via autonomous quantum thermal machines introduces thermodynamic neurons, where computation is performed through heat flows in a small quantum system coupled to baths at different temperatures. A central concept is the virtual qubit, which, together with a modulator, yields a perceptron-like architecture that can implement any linearly separable Boolean function; networks of such neurons can realize universal classical computation including XOR, via an algorithm that maps desired logic to energies and couplings. The framework remains thermodynamically consistent, enabling analysis of dissipation-accuracy trade-offs and offering a path toward analogue, physics-based neural networks. The work points to practical routes for experimental realization and raises questions about efficiency, autonomous learning, and the role of non-equilibrium dynamics in computation."

Abstract

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.

Figures (9)

• Figure 1: Thermodynamic neuron. The thermodynamic neuron is an autonomous quantum thermal machine designed for computing. The device consists of few interacting qubits (yellow dots), connected to several thermal environments. The input of the computation is encoded in the temperature of heat baths (depicted in red). This generates heat flows through the machine, which eventually reaches a non-equilibrium steady state. The output of the computation can be retrieved from the final temperature of a finite-size reservoir (shown in blue). By designing the machine (setting the qubit energies and their interaction), specific functions between the input and output temperatures can be implemented.
• Figure 2: Different operation regimes of a three-qubit thermal machine. The plot shows the inverse virtual temperature $\beta_v$ as a function of bath inverse temperature $\beta_1$ when keeping $\beta_0$ fixed. When $\beta_1 < \beta_0$, the inverse virtual temperature becomes larger than both $\beta_0$ and $\beta_1$, which means that the machine operates as a refrigerator. When $\beta_0 < \beta_1 < (\epsilon_0/\epsilon_1)\beta_0$ we have the exactly opposite situation and the machine operates as a heat pump. Finally, when $\beta_1 > (\epsilon_0/\epsilon_1)\beta_0$ the machine operates as a heat engine. Figure adapted from Ref. Brunner2012.
• Figure 3: Thermodynamic neuron for implementing a NOT gate. Panel A shows the design of the machine. The collector consists of three interacting qubits (yellow dots), each connected to a thermal environment. The logical input is encoded in the temperature $\beta_1$ of the heat bath $\mathcal{B}_1$ (red) while the output will be retrieved from the final temperature $\beta_z^{\infty}$ of the finite-size reservoir $\mathcal{B}_z$ (blue); the heat bath $\mathcal{B}_0$ is at fixed reference temperature. The collector implements the desired inversion of the temperature. To make the response non-linear, we must add the modulator, which consists of an additional qubit connected to a reference heat bath. Panel B shows the relation between the input temperature $\beta_1$ and the final output temperature $\beta_z^\infty$ (in the steady-state regime). Notably, the machine produces the desired inversion of the temperature. The quality of the response can be increased by tuning the machine parameters, in particular by increasing the energy gap $\epsilon_1$ of the collector qubit $\mathcal{C}_1$. Dashed black line shows the characteristics of an ideal NOT gate. Panel C shows the trade-off between the average dissipation $\langle \Sigma \rangle$ [see Eq. (\ref{['eq:avg_dissip']})] and the average error $\langle \xi \rangle$ [see Eq. (\ref{['eq:avg_error']})]. We see clearly that in order to increase robustness to noise, the machine must dissipate more heat to the environment. The inset shows the entropy production as a function of the input temperature $\beta_1$ for different values of the qubit energy $\epsilon_1$. Parameter values: $\beta_{\text{hot}} = 1$, $\beta_{\text{cold}} = 2$ , $\gamma = \chi = 1$, $\mu = 10^{-4}$, $\epsilon_z = 0.1$, $\tau = 10^8$ and $\beta_0 = \beta_z(0) = 3/2$.
• Figure 4: Virtual qubit in the collector. The sketch shows the energy structure of a three-qubit collector. The Hilbert space of the two physical qubits $\mathcal{C}_0$ and $\mathcal{C}_1$ contains a two-dimensional subspace with an energy gap $\epsilon_v = \epsilon_z$ (so-called virtual qubit) and effective temperature $\beta_v$ (so-called virtual temperature). The interaction Hamiltonian $H_{\text{int}}$ is chosen so that this virtual qubit interacts with the physical qubit $\mathcal{C}_z$, cooling it down (or heating up) in the process.
• Figure 5: General model of the thermodynamic neuron and analogy with a perceptron. Panel A shows the structure of a thermodynamic neuron for implementing an $n$-to-one bit function. The collector $\mathcal{C}$ consists of $n+2$ qubits, connected to the input heat baths (red), reference heat baths (grey), as well as the output reservoir (blue). The working principle of the collector is to thermalize qubit $\mathcal{C}_z$ to the virtual temperature $\beta_v$ (see Eq. (29)). In turn, this affects the temperature of the finite-size output reservoir $\mathcal{B}_z$ (blue). The modulator controls the range of output temperatures, making the response effectively non-linear. In the steady-state regime, the final output temperature is given by $\beta_z^{\infty}$ given by a non-linear function of $\beta_v$ [see Eqs \ref{['eq:beta_z_neuron']} and \ref{['eq:tn_expan']}]. The machine can implement any linearly-separable binary function by appropriately setting the parameters: the qubit energies, the interaction Hamiltonian and the temperatures of the reference heat baths. Notably, this machine is closely connected to the perceptron model shown in panel B, which is used extensively in machine learning. Given inputs $x_k$, the perceptron first computes a weighted sum $y$, then processed via a non-linear activation (sigmoid) function $f$. Similarly, the thermodynamic neuron first creates a virtual qubit at temperature $\beta_v$, which is a weighted sum of the input temperatures $\beta_k$. Second, the modulator implements the non-linear activation function. Note that in a specific regime ($\epsilon_z$ sufficiently small), the thermodynamic neuron implements a perceptron, as the activation function tends to a sigmoid in this case.