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Quantum Equilibrium Propagation for efficient training of quantum systems based on Onsager reciprocity
Clara C. Wanjura, Florian Marquardt
TL;DR
Physical gradient extraction for training remains challenging because broadly applicable approaches exist only for certain systems. This paper derives quantum equilibrium propagation from Onsager reciprocity and demonstrates it on quantum learning tasks. The examples include phase recognition and optimization trajectories that converge to phase boundaries.
Problem
Physically extracting training gradients remains challenging because general approaches apply only to certain physical systems.
Source: §I · p. 1Method
The paper derives quantum equilibrium propagation from Onsager reciprocity, using output nudges and measured responses to obtain gradients for quantum systems.
Source: §I · p. 2, B. Onsager reciprocity as a basis for… · p. 3Results
A two-qubit sensor learned the known cluster Ising phase diagram, while unsupervised trajectories converged to the gZZ = 0 line or a phase boundary.
Source: A. Supervised learning: phase… · p. 5, B. Unsupervised learning · p. 6Takeaways & Limitations
The authors propose QEP for quantum learning tasks on systems that are classically hard or impossible to simulate, including partially known or accessible Hamiltonians.
Source: §I · p. 2
Abstract
from arXiv · showhide
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.
I. INTRODUCTION
Neuromorphic systems promise more efficient physical computation, but training still lacks broadly applicable ways to extract gradients efficiently. The paper derives quantum equilibrium propagation from Onsager reciprocity and illustrates its use in quantum learning tasks.
I. INTRODUCTION
- Neuromorphic computing is motivated by rising energy consumption and training costs as neural networks grow in complexity.The field explores physical analogue neural networks across different platforms.Source: §I · p. 1
- Neuromorphic training remains challenging because physical gradient extraction is available only for certain cases, while feedback-based parameter shifting scales unfavourably with network size.Other approaches are specialized to particular nonlinearities, linear components, or sequential information processing.Source: §I · p. 1
- QEP’s output-nudging procedure requires one response experiment, or a small number when non-commuting observables must be measured, rather than experiments scaling linearly with parameter count.This comparison is between QEP and shifting each parameter separately in the parameter-shift method.Source: Figure 1 · p. 1
- Existing general physical backpropagation approaches apply to lossless systems with time reversal or to energy-based systems that equilibrate.These are Hamiltonian Echo Backpropagation and equilibrium propagation, respectively.Source: §I · p. 1
- Quantum equilibrium propagation (QEP) derives a quantum training procedure from the direct connection between equilibrium propagation and Onsager reciprocity.The approach is intended for quantum systems, including tuneable quantum many-body systems realized in quantum simulators.Source: §I · p. 2
- QEP is illustrated through supervised and unsupervised quantum tasks, including many-body phase recognition, phase-diagram exploration, sensing, and phase-boundary exploration.The authors propose that QEP could support tasks involving classically hard or impossible to simulate systems and partially known or accessible Hamiltonians.Source: §I · p. 2
A. Onsager reciprocity
For the quantum equilibrium states considered, Onsager reciprocity equates cross-responses: changing one parameter and observing another operator has the same susceptibility as exchanging their roles.
A. Onsager reciprocity
- A small static force coupled to operator A_j changes its expectation value through the ground-state linear response.The force enters the Hamiltonian as a term δλ_j A_j.Source: A. Onsager reciprocity · p. 2
- Onsager reciprocity makes the cross-susceptibility symmetric, so exchanging the perturbed parameter and measured operator gives the same response.The equation is stated in the preceding section as equality of the two cross-derivatives.Source: A. Onsager reciprocity · p. 2
- The reciprocity relation also applies to thermal states and, with a density matrix over degenerate ground states, to ground-state degeneracy.The degenerate-state density matrix is described as approximately equivalent to a thermal state at small nonzero temperature.Source: A. Onsager reciprocity · p. 2
B. Onsager reciprocity as a basis for quantum equilibrium propagation
Quantum equilibrium propagation uses Onsager reciprocity to obtain training gradients by nudging output degrees of freedom and measuring their effect on trainable degrees of freedom. This replaces parameter-by-parameter response experiments with a single error-signal experiment.
B. Onsager reciprocity as a basis for quantum equilibrium propagation
- The construction divides operators into input, trainable, and output sets, with input parameters fixed and output couplings set to zero during inference.The general QEP Hamiltonian is written as H(x, θ, ν).Source: B. Onsager reciprocity as a basis for… · p. 2
- Output observables may be projectors, making their expectation values outcome probabilities useful for classification.The output is read as expectation values of operators in the output set.Source: B. Onsager reciprocity as a basis for… · p. 3
- Instead of measuring each trainable parameter’s effect on the outputs, reciprocity lets an output force reveal the reciprocal responses of trainable degrees of freedom.Direct evaluation of all trainable-parameter susceptibilities requires experiments scaling linearly with their number.Source: B. Onsager reciprocity as a basis for… · p. 3
- A force vector ν = ∂L/∂y applied to the output degrees of freedom yields the loss gradient in a single experiment.This vector is the error signal; the measured response provides the susceptibilities needed for the training gradient.Source: B. Onsager reciprocity as a basis for… · p. 3
- The reciprocity-based translation of classical equilibrium propagation applies to quantum Hamiltonians of arbitrary structure.Source: B. Onsager reciprocity as a basis for… · p. 3
C. QEP procedure
QEP estimates supervised-learning gradients by measuring a system in free and nudged ground states. Onsager reciprocity links the output response to trainable-operator changes, enabling parameter updates from the measured expectation values.
C. QEP procedure
- For mean-square error, the error signal is ε = 2(y − ytarget(x)).Source: C. QEP procedure · p. 3
- QEP measures trainable-operator expectation values in free and nudged ground states, then uses Onsager reciprocity to approximate the loss gradient for parameter updates.The nudged Hamiltonian uses output forces ν = βε, where ε is the loss error signal.Source: C. QEP procedure · p. 3
- Positive and negative nudging can be combined to estimate gradients; this variant empirically performs better for finite nudging.Source: C. QEP procedure · p. 3
D. Practical requirements
Practical QEP requires tuneable couplings, output forces, measurable observables, and an efficient route to equilibrium. The paper describes both hybrid and autonomous approaches to ground-state preparation.
D. Practical requirements
- QEP requires scalable tuneable couplings, with examples including ion chains, superconducting-qubit arrays, neutral-atom Rydberg arrays, and atoms in optical lattices.Platforms operating out of equilibrium are not directly suitable unless they can be mapped to an equilibrium situation.Source: D. Practical requirements · p. 3
- Platforms must apply output forces and measure both output observables and trainable-coupling operators.Interaction expectations can be obtained from individual spin measurements; a Heisenberg-type coupling requires three component measurements or a collective measurement.Source: D. Practical requirements · pp. 3–4
- QEP requires efficient access to the ground state, achievable through hybrid approaches such as variational quantum eigensolvers or purely autonomous equilibration.The authors note that hybrid ground-state preparation may still be advantageous when the quantum many-body ground state is used efficiently.Source: D. Practical requirements · p. 4
A. Supervised learning: phase detection in a quantum many-body system
The section demonstrates supervised phase recognition using a trainable sensor coupled to a quantum many-body system. A two-qubit sensor learns the cluster-Ising phase diagram, while training must contend with projection shot noise and the trade-off introduced by finite nudging.
A. Supervised learning: phase detection in a quantum many-body system
- FIG. 2 tests phase recognition across the phase diagram and compares accuracy distributions for restricted couplings or training patches with the broader training setup.The caption specifies a chain of length N = 8 for the accuracy-evolution panel and M = 10 measurement shots per sample there.Source: Figure 2 · p. 5
- A two-qubit sensor coupled to two chain spins learns the cluster-Ising phase diagram with sufficient expressivity, and allowing non-commuting couplings yields good accuracy.The learned sensor-system couplings are not weak and visibly perturb nearby spins without deteriorating functionality.Source: A. Supervised learning: phase… · p. 5
- Projection shot noise makes gradient estimates noisy, with each measurement shot requiring renewed equilibration.Source: A. Supervised learning: phase… · p. 5
- Increasing finite nudging strength can counteract shot noise, but its deviation from linear response creates a trade-off for training convergence.The authors report a sweet spot balancing noise and nonlinearity; in their numerical experiments, small M with more batches best minimizes total experimental runs.Source: A. Supervised learning: phase… · p. 5
B. Unsupervised learning
The section presents QEP for unsupervised phase exploration and sensitivity optimization. In a cluster Ising model, optimization trajectories reach regimes of interest, including a phase boundary where an observable’s slope is largest.
Phase exploration
- Phase exploration optimizes a chosen expectation value from an initial point in the phase diagram, using QEP gradients to seek a potentially local maximum.For the example loss L = −y, the error signal is −1.Source: B. Unsupervised learning · pp. 5–6
- Maximizing the Néel order parameter ⟨X0 X4⟩ drives both example trajectories to gZZ = 0 and then into the paramagnetic phase.The examples fix gZXZ = −0.5; the loss decreases rapidly at first and more slowly along gZZ = 0.Source: B. Unsupervised learning · p. 6, Figure 3 · p. 6
- The authors suggest QEP could explore higher-dimensional phase diagrams that would take considerable effort to map by sweeping parameters.This is proposed for systems that may be partially unknown or computationally hard to simulate classically.Source: B. Unsupervised learning · p. 6
Sensitivity optimization
- Sensitivity optimization starts from two phase-diagram points and maximizes the difference-quotient slope of an observable with respect to a parameter.The authors suggest this could identify optimal sensor working points, such as for magnetic-field sensing.Source: B. Unsupervised learning · p. 6
- The optimized slope of ⟨X0 X4⟩ with respect to gX draws trajectories to a phase boundary, and one run follows the boundary.The authors suggest this behavior could help map phase boundaries.Source: B. Unsupervised learning · p. 6, Figure 3 · p. 6
IV. CONCLUSION AND OUTLOOK
The paper derives quantum equilibrium propagation (QEP) by exploiting Onsager reciprocity. It reports successful use in supervised learning with quantum-state inputs and in unsupervised learning tasks.
IV. CONCLUSION AND OUTLOOK
- The authors exploit Onsager reciprocity to derive a quantum version of equilibrium propagation.The passage describes classical EP as a general training technique for neuromorphic platforms.Source: §IV · p. 7
- The authors report successful QEP use when the input is effectively a quantum state, as in supervised classification of quantum phases.Source: §IV · p. 7
- The conclusion also reports successful use of QEP for unsupervised learning tasks.Source: §IV · p. 7
APPENDIX
The appendix details the supervised-learning setup and evaluation, motivates linear Hamiltonian nudging, and gives implementation details for phase exploration and sensitivity optimization.
Appendix: Some details for the supervised learning example
- The supervised example uses an 8-spin cluster Ising chain and a two-qubit sensor, with three projector outputs assigned one-hot phase labels.The coupled system has a 1024-dimensional Hilbert space; gradients use symmetric nudging and sparse Lanczos diagonalization, followed by Adam parameter updates.Source: Appendix: Some details for the… · p. 8
- Test accuracy is evaluated on 200 fixed, uniformly distributed points using either multiple projector expectations or a single sensor measurement.The appendix models shot noise with Gaussian draws of variance Var Aℓ/M, a reasonable approximation unless M is very small; repeated shots can boost single-shot accuracy.Source: Appendix: Some details for the… · p. 9
Appendix: Different approaches for nudging
- For nudging, the authors add linear output-operator terms with force νℓ = β∂L/∂yℓ rather than the full βL operator expression.They state higher-order terms are unnecessary for the gradient, may be sizable with quantum fluctuations, and could be harder to implement experimentally.Source: Appendix: Different approaches for… · p. 9
Phase exploration
- The phase-exploration example optimizes ⟨X0 X4⟩ in a 10-spin cluster Ising model, using sparse Lanczos ground states and learning rate and nudge parameter 0.1.The two illustrative starting points are gX = −0.1, gZZ = 0.4 and gX = 0.9, gZZ = 0.9.Source: Phase exploration · p. 9
Sensitivity optimization
- Sensitivity optimization targets the slope of ⟨X0 X4⟩ with respect to gX in the 10-spin cluster Ising model.One described setting trains gZZ while holding gZXZ = −0.5 constant; the derivative is approximated by comparing free and nudged expectation values.Source: Sensitivity optimization · pp. 9–10