Equilibrium Propagation: the Quantum and the Thermal Cases

TL;DR

This work generalizes Equilibrium Propagation (EP) to two new regimes: quantum and finite temperature. In the quantum extension, a quantum network is prepared in an energy eigenstate and trained via gradients derived from ground-state expectational values using a quantum variational framework. In the finite-temperature extension, EP leverages thermal fluctuations, deriving gradients from covariances with the cost under a Boltzmann distribution, which can remove the need for output clamping in training. The paper provides explicit gradient expressions, discusses measurement strategies (e.g., for $\hat X_i\hat X_j$), and develops a low-temperature expansion that yields deterministic learning equations using the Hessian. These contributions open pathways for implementing EP on quantum hardware and neuromorphic platforms operating under thermal noise, with future work needed on numerical performance and state-preparation strategies.

Abstract

Equilibrium propagation is a recently introduced method to use and train artificial neural networks in which the network is at the minimum (more generally extremum) of an energy functional. Equilibrium propagation has shown good performance on a number of benchmark tasks. Here we extend equilibrium propagation in two directions. First we show that there is a natural quantum generalization of equilibrium propagation in which a quantum neural network is taken to be in the ground state (more generally any eigenstate) of the network Hamiltonian, with a similar training mechanism that exploits the fact that the mean energy is extremal on eigenstates. Second we extend the analysis of equilibrium propagation at finite temperature, showing that thermal fluctuations allow one to naturally train the network without having to clamp the output layer during training. We also study the low temperature limit of equilibrium propagation.

Figures (1)

• Figure 1: In Equilibrium Propagation (EP), a neural network, with dynamic variables $z_i$ (represented by black dots) and internal parameters $\{ W_{ij} \}$, is operated at the minimum of an energy functional. During training the input and output layers ($I_{in}$ and $I_{out}$) are clamped to the vectors of the training set ($\{u_i\}$ and $\{ d_i\}$), and the response of the network to the clamping of the output is used to implement gradient descent on the internal parameters $\{ W_{ij} \}$. During normal operations, only the input is clamped, and the response of the output variables is the output of the network. In the quantum version described in Sec. \ref{['Sec:QEP']}, $z_i$ are operators, while $u_i$ and $d_i$ are external forces (for instance magnetic fields). Note that because the network is in equilibrium, information flows through the network in both directions: information on the $u_i$ and $d_i$ is spread out throughout the network. The Equilibrium Propagation method is suitable for general types of topologies (not only multi-layer networks).