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

TL;DR

This work introduces Quantum Equilibrium Propagation (QEP), a gradient-based training method for quantum systems derived from Onsager reciprocity to obtain parameter-gradients from a single nudged experiment. By partitioning a parameterized Hamiltonian $\hat{H}(x,\theta,\nu)$ into input, training, and output couplings and employing a free phase ($\nu=0$) and a nudged phase ($\nu=β\varepsilon$), QEP computes $\partial/\partial \theta_j \mathcal{L}$ via $\partial/\partial \theta_j \mathcal{L} \approx (\langle \hat{A}_j\rangle|_{\nu=β\varepsilon}-\langle \hat{A}_j\rangle|_{\nu=0})/\beta$. The paper demonstrates supervised learning for quantum phase detection using a small sensor coupled to a cluster Ising chain, and uncovers unsupervised tasks such as phase exploration and sensitivity optimization, with clear guidance on practical requirements for implementation on ion chains, superconducting qubits, neutral-atom arrays, and optical lattices. Importantly, QEP remains effective even when the Hamiltonian is partially unknown or classically intractable, and it can leverage ground-state preparation via hybrid (e.g., VQE) or autonomous schemes. The results illustrate a versatile, platform-agnostic route to quantum-enabled learning and phase-discovery tasks, with potential impact on quantum sensing and quantum simulation—paving the way for energy-efficient, physics-informed training on real quantum devices.

Abstract

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.

Figures (3)

• Figure 1: The concept of quantum equilibrium propagation. The goal is to efficiently and in a physical way obtain the gradient of some loss function (depending on expectation values measured at the "output" degrees of freedom of a quantum system) with respect to tuneable parameters. a Rather than shifting $N$ parameters separately and measuring the output expectation value for each shift (parameter-shift method), Onsager reciprocity dictates that the same information can be extracted by b shifting, i.e. nudging, only the parameters coupling to the output observables and (in a single go) measuring the response of all $N$ operators coupled to the training parameters (quantum equilibrium propagation). The latter procedure is more efficient as it requires only a single response experiment (or at most a small number of order 1, when some non-commuting observables have to be measured), whereas the parameter shift method requires a number of experiments scaling linearly with the number of parameters.
• Figure 2: Supervised learning: Learning recognition of quantum phases using QEP. (a) Schematic of a trainable quantum sensor coupled to a system. (b) Specific example of a two-qubit sensor coupled to a 1D cluster transverse Ising Hamiltonian at two locations, where readout of ${\hat{Z}}_{1'}$ and ${\hat{Z}}_{2'}$ is supposed to indicate the phase. The 51 tuneable couplings are learned using QEP. (c) Evolution of test accuracy during supervised training on the whole phase diagram (for a chain of length $N=8$). Multiple training runs (yellow), confidence intervals as areas (red; at 50% and 80%), and average accuracy (blue). Accuracies for "many queries" (asking whether the maximum-probability detector outcome matches correct phase) and "single shot" (probability to indicate correct phase in single quantum measurement) [batches of 10 training samples; projection noise for $M=10$ measurement shots per sample is accounted for; nudge parameter $\beta=0.4$]. (d) Overlap of batch-averaged gradient estimate with true gradient direction, vs. nudge parameter. Confidence intervals (red, 95%, 80%, 50%) depict distribution over many batches (batch size 10, no measurement shot noise). (e) Gradient overlap histograms vs. measurement shots $M$, for two different nudge parameters (batch size 10). (f) Test of phase recognition: probabilities of measuring the trained sensor in one of the three different combinations of $Z_{1'},Z_{2'}$ shown in orange/green/blue; true phase boundaries in black ($g_{\rm Z}+g_{\rm XX}+g_{\rm ZXZ}=4$). (g) Histogram of final test accuracies for repeated training runs (parameters as above), for a sensor that only couples to ${\hat{Z}}$ operators in the chain (or only to ${\hat{X}}$ and ${\hat{Z}}$), and for a sensor trained only on a small patch in the middle of each phase but tested throughout ("restricted").
• Figure 3: Unsupervised learning applications: phase exploration and sensitivity optimization for sensing.a Sketch of phase space exploration using QEP. An expectation value of interest, $\langle \hat{A}_\ell\rangle$, is maximised. b Two example trajectories in the phase diagram of the cluster Ising Hamiltonian \ref{['eq:ClusterIsing']} each starting at the position in the phase diagram marked with $x$ and c the corresponding loss functions. The colours in b show $\langle {\hat{X}}_0 {\hat{X}}_4\rangle$. d Sketch of the concept of sensitivity optimization. The derivative of a an observable of interest, $\langle {\hat{A}}_\ell\rangle$, w.r.t. a certain parameter (or w.r.t. a vector of parameters) is optimised. Concretely, this scheme can be used to devise optimal sensors or to find phase boundaries in a phase diagram. e-g Optimization of $\partial\langle {\hat{X}}_0 {\hat{X}}_4\rangle/\partial g_X$ (magnetic field sensor). e Two example runs, each showing the trajectories of two points in the phase diagram of the cluster Ising Hamiltonian \ref{['eq:ClusterIsing']} and f the corresponding loss functions. The two points are inialized at the position marked with $\times$ and converge towards the phase boundary where the derivative w.r.t. $g_X$ is maximal. In the second run, g the trajectories first converge towards a phase boundary and then follow it, suggesting that the technique may be employed to trace out phase boundaries. The plots on the right show cuts through the phase diagram at various steps during the training.