Exploiting quantum chaos diagnostics in QAOA for enhanced hybrid quantum classical deep learning classification

Abstract 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.