Foundations of Practical Quantum Advantage in Quantum-Informed Machine Learning for Predicting Chaos

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 n_q = 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 n_q qubits. In the extraction stage, joint Bell measurements on two copies estimate any post hoc Pauli functional with a copy-pair count independent of n_q, whereas any adaptive single-copy protocol for the corresponding full-Pauli read-out requires Omega(2^(n_q)) 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: a turbulent channel-flow study in which the two-copy read-out yields a named non-diagonal correlator of the invariant measure (the velocity-direction coherence), and a medium-range weather forecasting workflow on the European Centre for Medium-Range Weather Forecasts ERA5 reanalysis in which the diagonal k <= 2 Q-Prior steers a Koopman rollout, improves anomaly-correlation skill by 10-39% across 48-240 h lead times, and reduces the long-horizon collapse of rollouts onto a static mean field. The two conditions of our practical-advantage definition are met at complementary levels, identifying a candidate route to practical quantum advantage before fault-tolerant hardware.