⚛️ quantum physics

Equilibrium propagation in non-equilibrium quantum systems

This paper extends the equilibrium propagation training algorithm to driven-dissipative many-body quantum systems by deriving new non-equilibrium susceptibility relations that leverage quantum detailed balance symmetry, thereby enabling efficient learning without the need for repeated ground-state relaxation.

Original authors: Aditya Gandotra, David D. Noachtar, Aashish A. Clerk

Published 2026-10-27

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CC BY 4.0

Original authors: Aditya Gandotra, David D. Noachtar, Aashish A. Clerk

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

In the emerging field of physical learning, scientists explore how machines can learn by using the laws of physics themselves as the engine of computation. Instead of running software on a silicon chip, these systems use the natural behavior of atoms, light, or electrical circuits to process information. The goal is to teach a physical object to perform a specific task, like recognizing a pattern or solving a problem, by adjusting its internal settings. This process usually relies on a method called gradient descent, where the system makes tiny adjustments to its parameters to reduce errors, much like a hiker feeling their way down a mountain in the fog to find the lowest valley. For this to work efficiently, the system needs a reliable way to calculate how a small change in one setting affects the final result. In the world of standard physics, where things settle into a calm, stable state, there is a powerful shortcut known as equilibrium propagation that makes this calculation incredibly fast. However, this shortcut has historically required the system to be in a state of perfect thermal balance, a condition that is extremely difficult to maintain in complex quantum systems, especially when they are being actively driven and cooled by their environment.

Researchers at the University of Chicago have now found a way to bypass this limitation, showing that this efficient learning method can work even when a quantum system is far from equilibrium. Their work focuses on driven-dissipative systems, which are quantum objects that are constantly being pushed by an external force while simultaneously losing energy to their surroundings. In such systems, the object never settles into a quiet, static state; instead, it flows into a steady rhythm that depends on how hard it is being pushed. Traditionally, scientists believed that the efficient learning shortcut could not apply here because these flowing states do not have a simple energy map to guide them. The team demonstrated, however, that a specific hidden symmetry in the way these systems behave allows the shortcut to function perfectly. They proved that for certain types of quantum systems, the relationship between cause and effect remains predictable and symmetrical, even without a traditional energy landscape.

To test this idea, the researchers simulated two distinct types of quantum machines. The first was a network of tiny magnets, or spins, that interact with each other while being driven by external fields and losing energy. The second was a grid of light particles, or bosons, that interact and are driven by a different kind of force. In both cases, they set up a "teacher-student" scenario. The teacher was a system with a known set of settings that produced a specific, complex output. The student was an identical system, but with its internal settings initially set to random values. The goal was to train the student to mimic the teacher's behavior. The researchers used the new equilibrium propagation method to adjust the student's settings, comparing it against the older, more cumbersome method that requires resetting the system from scratch for every single calculation.

The results showed that the new method worked just as well as the old one in terms of accuracy, but it was significantly more efficient. In the simulations, the new approach required far fewer attempts to prepare the system in its steady state to reach the same level of learning. This is a crucial advantage because preparing these complex quantum states is often the most time-consuming and difficult part of the process. The researchers found that the efficiency gain was substantial, scaling in a way that suggests the method could handle much larger and more complex systems than previously thought possible. They also observed that while the student system eventually learned to produce the correct output, the internal settings it settled into were not always identical to the teacher's. This indicates that the system found a different internal path to the same result, a common feature in learning where multiple solutions can exist.

The key to this success was the discovery that these non-equilibrium systems possess a hidden symmetry, a mathematical property that ensures the system's response to changes remains consistent and reversible in a specific way. This symmetry acts as a substitute for the energy map that is missing in these active systems. By relying on this property, the researchers could derive a rule that allows the system to calculate the necessary adjustments for learning without needing to know the full energy state of the machine. This finding opens the door to using a much wider variety of physical systems for learning tasks, including those that are inherently active and dynamic rather than static and calm. It suggests that the potential for physical learning is not limited to systems that can be cooled to a perfect standstill, but extends to the rich, flowing world of driven quantum matter.

The study did not claim to have solved all the challenges of quantum learning, nor did it demonstrate these systems on actual hardware. The results were derived from precise numerical simulations that modeled the behavior of these systems with high accuracy. The researchers noted that while their specific models worked perfectly, the broader question of how expressive these systems are—meaning how complex a task they can learn—remains an open area for future investigation. They also pointed out that many real-world systems might only possess this helpful symmetry approximately, rather than perfectly. In such cases, the learning method might still work, though it could take longer to find the best solution. This possibility suggests that the approach could be robust enough to handle the imperfections of real experimental setups.

Ultimately, this work provides a new blueprint for training physical machines. It shows that the strict requirement for a system to be in thermal equilibrium is not a hard barrier for efficient learning. By identifying the specific symmetries that allow learning to proceed in driven, dissipative systems, the researchers have expanded the toolkit available for building physical computers. The ability to train these systems efficiently without needing to reset them constantly brings the vision of self-learning quantum devices closer to reality. The findings suggest that the future of physical learning may lie not in forcing quantum systems into a quiet state, but in harnessing their natural, active dynamics to perform complex tasks.

Technical Summary: Equilibrium Propagation in Non-Equilibrium Quantum Systems

Problem Statement
Equilibrium Propagation (EP) is a powerful training algorithm for physical learning systems that extracts gradients efficiently by exploiting the existence of an energy or free energy function. While successfully applied to classical systems and equilibrium quantum systems (where states are ground or thermal states), a significant bottleneck exists for quantum applications: EP requires the system to re-equilibrate to a ground state after every parameter update. Preparing ground states for many-body Hamiltonians is generally QMA-hard, and efficient preparation is often limited to specific regimes (e.g., high temperatures or 1D systems). This reliance on repeated ground state preparation poses a severe scalability challenge for large-scale quantum learning.

The authors address the question of whether EP can be extended to driven-dissipative quantum systems, which naturally relax to parameter-dependent non-equilibrium steady states (NESS) without requiring external cooling mechanisms. A direct application of EP fails here because these non-equilibrium states lack a well-defined free energy, and the standard susceptibility symmetries required for EP are not guaranteed.

Methodology and Theoretical Framework
The paper proposes extending EP to open quantum systems governed by Markovian Lindblad master equations. The core methodology relies on identifying a specific class of non-equilibrium systems that satisfy Hidden Time-Reversal Symmetry (hTRS), a form of quantum detailed balance.

Hidden Time-Reversal Symmetry (hTRS): The authors utilize the condition where the effective non-Hermitian Hamiltonian and jump operators satisfy specific constraints involving an anti-linear operator $\hat{\Psi}$ . While hTRS allows for NESS that are far from thermal Gibbs states, it imposes strong constraints on the system's dynamics.
Susceptibility Symmetries: The central theoretical insight is that hTRS promotes trivial gauge covariances (relating steady states under unitary transformations of drive phases) to non-trivial relationships that connect steady states with different drive magnitudes and phases. This leads to the derivation of static susceptibility symmetries in the NESS. Specifically, the authors prove that for certain observables, the response of an observable $i$ to a drive $j$ equals the response of observable $j$ to drive $i$ ( $\partial \langle \hat{O}_i \rangle / \partial \theta_j = \partial \langle \hat{O}_j \rangle / \partial \theta_i$ ), even in the absence of a free energy.
EP Training Protocol: The authors adapt the standard two-phase EP protocol:
- Free Phase: The system relaxes to the NESS $\hat{\rho}_0$ defined by current parameters.
- Nudged Phase: The system is perturbed by shifting the drives on output sites proportional to the error vector ( $\vec{\theta}' \to \vec{\theta}' + \beta \vec{\epsilon}$ ), and the system relaxes to a new NESS $\hat{\rho}_\beta$ .
- Gradient Estimation: The gradient is computed using the difference in expectation values between the nudged and free states, relying on the derived susceptibility symmetry to ensure correctness.

Key Contributions and Models
The paper derives these results for two experimentally relevant many-body models, demonstrating that hTRS enables exact susceptibility symmetries:

Dissipative Transverse-Field Ising Model (DTFIM): A system of $N$ Rabi-driven qubits with all-to-all Ising interactions and local relaxation. The authors show that for drive amplitudes $\Omega_j$ , the steady-state coherences $\langle \hat{\sigma}_j^y \rangle$ satisfy the required symmetry $\partial \langle \hat{\sigma}_i^y \rangle / \partial \Omega_j = \partial \langle \hat{\sigma}_j^y \rangle / \partial \Omega_i$ . Notably, the observable $\hat{\sigma}^y$ does not appear in the Hamiltonian, highlighting the non-equilibrium nature of the symmetry.
Parametrically-Driven Bosonic Lattice: A system of interacting bosonic modes with two-photon drives and Kerr nonlinearities. The authors establish that for real-valued drive amplitudes $M_{ij}$ , the steady-state pair correlations $\langle \hat{Y}_{ij} \rangle$ (where $\hat{Y}_{ij} \propto i(\hat{a}_i \hat{a}_j - \hat{a}_i^\dagger \hat{a}_j^\dagger)$ ) satisfy a generalized susceptibility symmetry. This model features a spatial structure and unbounded local Hilbert space, offering greater complexity than the DTFIM.

Results
The authors benchmarked the proposed EP method against the standard parameter-shift method using a "teacher-student" learning setup (supervised learning where a student system learns the input-output mapping of a teacher system).

DTFIM Simulation: For an $N=11$ spin system, EP converged to the same target input-output relation as the parameter-shift method. However, EP required significantly fewer state preparations. The authors report a speedup factor of approximately $2M_{train}/3 \approx 6.67$ in terms of the number of distinct steady-state preparations needed per gradient update.
Bosonic Lattice Simulation: For an $N=4$ mode system, EP similarly outperformed the parameter-shift method, showing a speedup factor of 6. The bosonic model exhibited non-monotonic, complex input-output relations, suggesting higher expressivity compared to the DTFIM.
Efficiency: In both cases, EP reduced the scaling of state preparations from linear in the number of parameters (required by parameter-shift) to a constant number (typically 2 or 3 states) per gradient update, independent of the number of trainable parameters.

Significance and Claims
The paper claims to establish a viable route for physical learning in non-equilibrium quantum systems without the prohibitive cost of repeated ground state preparation.

Theoretical Breakthrough: The work demonstrates that the "equilibrium" in Equilibrium Propagation does not strictly require thermal equilibrium or a free energy. Instead, it requires a specific symmetry of static susceptibilities, which can be guaranteed by hTRS in driven-dissipative systems.
Scalability: By leveraging the natural relaxation dynamics of open quantum systems, the proposed method circumvents the QMA-hardness of ground state preparation, offering a scalable training algorithm for many-body quantum systems.
Scope: The authors explicitly state that their focus is on trainability (the efficiency of the gradient descent algorithm) rather than expressivity (the capacity of the system to learn complex functions). While the results suggest that non-equilibrium systems with hTRS may possess rich expressive capabilities, a systematic study of this is left for future work.
Generality: The authors note that while they demonstrate this for two specific models, the approach could apply to other systems known to possess hTRS, such as dissipative spin chains and interacting fermionic models. They also suggest that systems with approximate hTRS could be utilized, potentially expanding the class of usable physical systems.

Technical Summary: Equilibrium Propagation in Non-Equilibrium Quantum Systems

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