# Equilibrium Propagation: the Quantum and the Thermal Cases

arXiv: https://arxiv.org/abs/2405.08467 · HTML: https://mlsanity.com/paper/2405.08467/

## Key facts

- Cited by 10 papers in the map (data through 14 Oct 2026).
- Builds on 11 of its 32 references that are in the map.
- First posted on arXiv 14 May 2024.

## Builds on (most cited first, 11 of 11)

- [Barren plateaus in quantum neural network training landscapes](https://mlsanity.com/paper/1803.11173/) (2018)
- [The power of quantum neural networks](https://mlsanity.com/paper/2011.00027/) (2020)
- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Challenges and Opportunities in Quantum Machine Learning](https://mlsanity.com/paper/2303.09491/) (2023)
- [Efficient Learning for Deep Quantum Neural Networks](https://mlsanity.com/paper/1902.10445/) (2019)
- [Scaling Equilibrium Propagation to Deep ConvNets by Drastically Reducing its Gradient Estimator Bias](https://mlsanity.com/paper/2006.03824/) (2020)
- [Holomorphic Equilibrium Propagation Computes Exact Gradients Through Finite Size Oscillations](https://mlsanity.com/paper/2209.00530/) (2022)
- [Updates of Equilibrium Prop Match Gradients of Backprop Through Time in an RNN with Static Input](https://mlsanity.com/paper/1905.13633/) (2019)
- [EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations](https://mlsanity.com/paper/2010.07859/) (2020)
- [Equilibrium Propagation with Continual Weight Updates](https://mlsanity.com/paper/2005.04168/) (2020)
- [Detailed Balanced Chemical Reaction Networks as Generalized Boltzmann Machines](https://mlsanity.com/paper/2205.06313/) (2022)

## Built on by (most cited first, 10 of 10)

- [Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems](https://mlsanity.com/paper/2406.00879/) (2024)
- [Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity](https://mlsanity.com/paper/2406.06482/) (2024)
- [Equilibrium Propagation for Learning in Lagrangian Dynamical Systems](https://mlsanity.com/paper/2505.07363/) (2025)
- [Near-Equilibrium Propagation training in nonlinear wave systems](https://mlsanity.com/paper/2510.16084/) (2025)
- [Equilibrium Propagation for Non-Conservative Systems](https://mlsanity.com/paper/2602.03670/) (2026)
- [Training a Predictive Coding Network on ImageNet using Equilibrium Propagation](https://mlsanity.com/paper/2606.03584/) (2026)
- [A Blueprint for Equilibrium-Based Differentiable Continuous-Variable Thermodynamic Computing](https://mlsanity.com/paper/2607.16183/) (2026)
- [Finite-Nudge Equilibrium Propagation in Thermal Ensembles](https://mlsanity.com/paper/2511.22024/) (2025)
- [Optical Implementation of Equilibrium Propagation Using Spatial Photonic Ising Machines](https://mlsanity.com/paper/2606.13454/) (2026)
- [Unifying Physical Backpropagation](https://mlsanity.com/paper/2608.11585/) (2026)

## Abstract (arXiv)

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.