# Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity

arXiv: https://arxiv.org/abs/2406.06482 · HTML: https://mlsanity.com/paper/2406.06482/

## Key facts

- Cited by 9 papers in the map (data through 14 Oct 2026).
- Builds on 24 of its 61 references that are in the map.
- First posted on arXiv 10 Jun 2024.

## Builds on (most cited first, 24 of 24)

- [Equilibrium Propagation: Bridging the Gap Between Energy-Based Models and Backpropagation](https://mlsanity.com/paper/1602.05179/) (2016)
- [Deep physical neural networks enabled by a backpropagation algorithm for arbitrary physical systems](https://mlsanity.com/paper/2104.13386/) (2021)
- [Photonics for artificial intelligence and neuromorphic computing](https://mlsanity.com/paper/2011.00111/) (2020)
- [Assessing the Scalability of Biologically-Motivated Deep Learning Algorithms and Architectures](https://mlsanity.com/paper/1807.04587/) (2018)
- [Training End-to-End Analog Neural Networks with Equilibrium Propagation](https://mlsanity.com/paper/2006.01981/) (2020)
- [Training of Physical Neural Networks](https://mlsanity.com/paper/2406.03372/) (2024)
- [Training of photonic neural networks through in situ backpropagation](https://mlsanity.com/paper/1805.09943/) (2018)
- [Experimentally realized in situ backpropagation for deep learning in nanophotonic neural networks](https://mlsanity.com/paper/2205.08501/) (2022)
- [Backpropagation-free Training of Deep Physical Neural Networks](https://mlsanity.com/paper/2304.11042/) (2023)
- [Energy-based learning algorithms for analog computing: a comparative study](https://mlsanity.com/paper/2312.15103/) (2023)
- [EqSpike: Spike-driven Equilibrium Propagation for Neuromorphic Implementations](https://mlsanity.com/paper/2010.07859/) (2020)
- [Training an Ising Machine with Equilibrium Propagation](https://mlsanity.com/paper/2305.18321/) (2023)
- [Self-learning Machines based on Hamiltonian Echo Backpropagation](https://mlsanity.com/paper/2103.04992/) (2021)
- [Equilibrium Propagation with Continual Weight Updates](https://mlsanity.com/paper/2005.04168/) (2020)
- [A deep learning theory for neural networks grounded in physics](https://mlsanity.com/paper/2103.09985/) (2021)
- [Nonlinear Processing with Linear Optics](https://mlsanity.com/paper/2307.08533/) (2023)
- [Machine Learning Without a Processor: Emergent Learning in a Nonlinear Electronic Metamaterial](https://mlsanity.com/paper/2311.00537/) (2023)
- [Training Coupled Phase Oscillators as a Neuromorphic Platform using Equilibrium Propagation](https://mlsanity.com/paper/2402.08579/) (2024)
- [Agnostic Physics-Driven Deep Learning](https://mlsanity.com/paper/2205.15021/) (2022)
- [Nonlinear optical encoding enabled by recurrent linear scattering](https://mlsanity.com/paper/2307.08558/) (2023)
- [Equilibrium Propagation: the Quantum and the Thermal Cases](https://mlsanity.com/paper/2405.08467/) (2024)
- [Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems](https://mlsanity.com/paper/2406.00879/) (2024)
- [Training neural networks with end-to-end optical backpropagation](https://mlsanity.com/paper/2308.05226/) (2023)
- [Local minima in quantum systems](https://mlsanity.com/paper/2309.16596/) (2023)

## Built on by (most cited first, 9 of 9)

- [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)
- [Circuit realization and hardware linearization of monotone operator equilibrium networks](https://mlsanity.com/paper/2509.13793/) (2025)
- [Laser interferometry as a robust neuromorphic platform for machine learning](https://mlsanity.com/paper/2601.18047/) (2026)
- [The Phase Is the Gradient: Equilibrium Propagation for Frequency Learning in Kuramoto Networks](https://mlsanity.com/paper/2604.10272/) (2026)
- [Equilibrium Propagation and Hamiltonian Inference in the Diffusive Fitzhugh-Nagumo Model](https://mlsanity.com/paper/2605.21568/) (2026)
- [Perturbative Contrastive Physical Learning](https://mlsanity.com/paper/2606.09756/) (2026)

## Abstract (arXiv)

The widespread adoption of machine learning and artificial intelligence in all branches of science and technology has created a need for energy-efficient, alternative hardware platforms. While such neuromorphic approaches have been proposed and realised for a wide range of platforms, physically extracting the gradients required for training remains challenging as generic approaches only exist in certain cases. Equilibrium propagation (EP) is such a procedure that has been introduced and applied to classical energy-based models which relax to an equilibrium. Here, we show a direct connection between EP and Onsager reciprocity and exploit this to derive a quantum version of EP. This can be used to optimize loss functions that depend on the expectation values of observables of an arbitrary quantum system. Specifically, we illustrate this new concept with supervised and unsupervised learning examples in which the input or the solvable task is of quantum mechanical nature, e.g., the recognition of quantum many-body ground states, quantum phase exploration, sensing and phase boundary exploration. We propose that in the future quantum EP may be used to solve tasks such as quantum phase discovery with a quantum simulator even for Hamiltonians which are numerically hard to simulate or even partially unknown. Our scheme is relevant for a variety of quantum simulation platforms such as ion chains, superconducting qubit arrays, neutral atom Rydberg tweezer arrays and strongly interacting atoms in optical lattices.