The Quantum Approximate Optimization Algorithm (QAOA) is repurposed here as a feature map within a hybrid quantum–classical classifier, augmented by a chaos-informed diagnostic. We extract a scalar chaos feature by evaluating an Out-Of-Time-Ordered correlators (OTOC) along parameter-scaling rays through the trained circuit, computing spacings between local minima, and standardizing them via a pre-fitted lognormal model. To probe finite-size effects, we sweep the number of qubits \(n\in \{4,6,8,10\}\) at fixed depth \(p=2\) and train two models on a balanced 1,000-sample MNIST subset: a StandardHybrid using the \(n\) local Pauli- \(Z\) expectations, and a ChaosAwareHybrid which appends the OTOC-derived scalar. We perform multi-run, 5-fold cross-validation with a paired design (identical seeds/folds across models) and report mean±SD, paired mean differences \(\Delta\) , 95% t- and bootstrap CIs, exact permutation/sign tests, win-rates (Wilson 95% CI), and paired effect sizes. Across \(N_\text {pairs}=\{50,50,67,50\}\) for \(n=\{4,6,8,10\}\) , the chaos-aware variant significantly improves test accuracy at \(n\in \{4,6,8\}\) with \(\Delta \approx +0.016\) – \(+0.018\) , all 95% CIs excluding zero, permutation \(p\approx 0\) , high win-rates (86–100%), and large paired effects ( \(d_z\approx 1.0\) –2.3). At \(n=10\) the effect reverses ( \(\Delta =-0.022\) , 2% win-rate, \(d_z=-2.20\) ), indicating over-sensitivity. The best average accuracy occurs at \(n=8\) ( \(0.9006\pm 0.0069\) ; \(\Delta =+0.0180\) ; 100% wins). Per-epoch panels (train/val/test; mean±1 SD) reveal a “Goldilocks” width at which expressivity and sensitivity are balanced. These results show that a calibrated chaos diagnostic can enhance hybrid quantum–classical classifiers in resource-limited regimes and provide a principled knob to match circuit expressivity to many-body sensitivity.
