Foundations of Practical Quantum Advantage in Quantum-Informed Machine Learning for Predicting Chaos
Authors: Maida Wang, Xiao Xue, Minh Chung, Peter V. Coveney
Organizations: Centre for Computational Science, University College London, London, UK · Leibniz Supercomputing Centre of the Bavarian Academy of Sciences and Humanities, Boltzmannstrasse 1, 85748 Garching, Germany · Centre for Advanced Research Computing, University College London, London, UK
We develop theoretical foundations for a practical quantum-advantage mechanism in quantum-informed machine learning for chaotic dynamical systems. A family of k-indexed higher-order quantum statistical priors (Q-Priors) hosts the k-point marginal of the invariant measure on nq=kq qubits, extending the single-site construction of prior work. We prove a two-stage advantage. In the representation stage, superposition and entanglement compactly store non-factorisable spatial correlations of the invariant measure on nq qubits. In the extraction stage, joint Bell measurements on two copies estimate any \emph{post hoc} Pauli functional with a copy-pair count independent of nq, whereas any adaptive single-copy protocol for the corresponding full-Pauli read-out requires Ω(2nq) copies; this is a provable quantum-classical separation in copy-measurement complexity. The two-copy read-out is realised in simulation and on IQM superconducting processors. Two case studies instantiate the mechanism in workflows of independent scientific value. In a turbulent channel-flow study, the two-copy read-out yields the velocity-direction coherence as a named non-diagonal correlator of the invariant measure, and the multi-site k=2 Q-Prior recovers DNS-level invariant-measure statistics that the unregularised baseline loses. In a medium-range weather forecasting workflow on the European Centre for Medium-Range Weather Forecasts ERA5 reanalysis, the diagonal k≤2 Q-Prior steers a Koopman rollout, improves anomaly correlation skill by 10% to 39% across 48 to 240h lead times, and stabilises long-horizon rollouts against collapse onto a static mean field. Together, the mechanism and these two case studies satisfy our practical-advantage definition, identifying a candidate route to practical quantum advantage before fault-tolerant hardware.
Machine-learning surrogates of physical systems face a paradox: explainable models facing the challenge of expressivity to capture complex nonlinear flows, whereas expressive deep surrogates match high-fidelity simulations only through massive parameterisations that turn the learned dynamics into a black box. Here, we introduce quantum-compressed machine learning (QCML), which resolves this tension by compressing the latent propagator of a flow surrogate from 524,288 trainable parameters to no more than 8. This parameter reduction brings the learned dynamical law to the parameter scale of a physical constitutive relation rather than a black-box neural network, making the surrogate directly interpretable and controllable without sacrificing expressivity. The compression is realised by a structured quantum circuit whose unitary propagator constrains the latent spectrum to the unit circle exactly and by construction, replacing exponential error growth with linear accumulation over autoregressive rollouts. Classical regularisation only approximates this constraint: even a quantum-inspired classical baseline penalised towards unitarity collapses within one Lyapunov time on turbulent channel flow, whereas QCML remains stable over the full rollout. Shared phase and coupling angles parameterising the circuit correspond directly to modal frequencies and inter-mode interactions, giving the learned dynamics a physical interpretation in spectral space. On two patient-specific cardiovascular benchmarks, the structured QCML propagator matches the predictive accuracy of its classical counterpart on surface pressure spectra, pressure drop, and wall shear stress. These results establish QCML as a working component of scientific machine learning and a concrete contribution towards practical quantum advantage in real-world prediction.
Quantum reservoir computing uses a fixed quantum circuit as a feature generator and trains only a simple linear readout on top of it. This makes it cheap to train and free of the optimisation problems that affect many quantum machine-learning models. A natural worry is that the very large feature space the circuit produces might inflate apparent performance without adding anything real. This paper provides two things. First, it gives a complete, reproducible recipe for one such reservoir applied to forecasting chaotic systems, including how data is fed in, how the circuit is built, and how the readout is trained. Second, it gives a way to tell whether the reservoir's high dimension is actually doing useful work. We grow the size of the prediction problem and the size of the quantum reservoir together, so that extra capacity cannot be the explanation for any improvement, and we track a single stability number that measures how well behaved the readout fit is. On two chaotic test systems, a spatiotemporal chain and a shallow-water fluid model, the quantum reservoir keeps a flat, stable error as both sizes grow, while a matched classical reservoir does not. We report where the classical baseline is in fact stronger, so the comparison is honest. The result is a clean specification plus a diagnostic that other groups can apply to any reservoir whose features have a known scale.
Quantum Machine Learning (QML) has shown rapid advances by utilizing quantum computing for machine learning tasks. Meanwhile, the privacy risks accompanying QML is also starting to be studied, which inherit privacy leakage channels from "classical" ML and also quantum-unique risks. Existing work on privacy-preserving QML largely focuses on a QML-as-a-service scenario, which generally assumes that the QML model owner provides only classical bit outputs to queries, while users (and adversaries) have only classical computing abilities. However, this view is increasingly challenged in a quantum-native world of quantum-capable users/adversaries, which may have access to both quantum computing abilities and access to quantum information output from service providers. In this paper, we aim to bridge this gap by examining membership inference attacks against QML models by demonstrating that increasing quantum access and quantum computing abilities provides provable theoretical privacy leakage and empirical adversarial gain. However, the probabilistic nature of QML introduces a gap between theoretical and empirical adversarial advantage. These results show that existing research on privacy leakage in QML models underestimates privacy leakage in emergent quantum-native access regimes, and we hope to establish a first step in examining potential privacy leakages for QML in the quantum-native world.