Quantum Equilibrium Propagation: Gradient-Descent Training of Quantum Systems

TL;DR

The paper addresses how to perform gradient-based training of quantum systems using a physics-grounded framework. It generalizes Equilibrium Propagation (EP) to the quantum domain as Quantum Equilibrium Propagation (QEP), where the ground state of a parameterized Hamiltonian minimizes the mean energy, and gradients of a cost observable are obtained from two nudged equilibria via the total Hamiltonian $\\hat H^\\beta = \\hat H + \\beta \\widehat{C}$. It formalizes a local learning rule, analyzes gradient estimation under quantum measurement—including multiple measurements and non-commuting observables—and demonstrates QEP on the transverse-field Ising model and a quantum harmonic oscillator network. The work highlights locality of updates, potential hardware advantages for energy-efficient quantum ML, and connections to recent quantum EP developments, while outlining practical challenges and future directions for scaling and theory (e.g., holomorphic EP and agnostic EP). This establishes a principled, physically grounded route to train quantum systems via energy-based objectives and gradient-based optimization.

Abstract

Equilibrium propagation (EP) is a training framework for energy-based systems, i.e. systems whose physics minimizes an energy function. EP has been explored in various classical physical systems such as resistor networks, elastic networks, the classical Ising model and coupled phase oscillators. A key advantage of EP is that it achieves gradient descent on a cost function using the physics of the system to extract the weight gradients, making it a candidate for the development of energy-efficient processors for machine learning. We extend EP to quantum systems, where the energy function that is minimized is the mean energy functional (expectation value of the Hamiltonian), whose minimum is the ground state of the Hamiltonian. As examples, we study the settings of the transverse-field Ising model and the quantum harmonic oscillator network -- quantum analogues of the Ising model and elastic network.

Theorems & Definitions (7)

• Theorem 1: Equilibrium Propagation
• Lemma 1: Variational formulation of the ground state
• Theorem 2: Quantum Equilibrium Propagation
• Lemma 2: Variational formulation of the ground state
• proof : Proof of Lemma \ref{['lma:variational-formulation']}
• Theorem 2: Quantum Equilibrium Propagation
• proof : Proof of Theorem \ref{['thm:quantum-equilibrium-propagation']}