Quantum Feature Selection with Higher-Order Binary Optimization on Trapped-Ion Hardware

Credit default prediction is a tabular classification problem in which modest gains in F1 translate directly into reduced financial exposure. We ask whether Instantaneous Quantum Polynomial-time (IQP) circuits can produce features that improve a classifier over both its raw classical baseline and Kernel PCA - the strongest unsupervised classical non-linear alternative - at an equal feature budget. The dataset provides 23 financial attributes per client; for an n-qubit circuit we select n of them, encode each as a rotation angle, and read 2n expectation values back out as new features. The motivation for using a quantum circuit is computational: an n-qubit IQP circuit runs in constant depth and encodes feature correlations in a 2^n-dimensional Hilbert space, whereas classical simulation of its exact output statistics scales exponentially in n. Using the UCI Default of Credit Card Clients dataset and five-fold cross-validation, we find that appending 16 IQP features (n = 8 qubits) to a Logistic Regression model raises F1 from 0.462 to 0.517 (+0.055, p < 0.0001). Kernel PCA, the next-best method, reaches only 0.493 at the same feature count; the gap survives Benjamini-Hochberg correction across 12 tests (p = 0.00007). No other classifier - Random Forest, SVM, XGBoost, or k-NN - benefits, which points to a linear-expressivity mechanism rather than a generic improvement. We also show that how the 8 input features are chosen matters: Random Forest importance-guided selection reaches F1 = 0.523, while encoding maximally uncorrelated features drops it to 0.496, demonstrating that the circuit amplifies informative structure rather than creating it from scratch.

Training quantum neural networks (QNNs) on quantum hardware is currently bottlenecked by the cost of gradient estimation: standard parameter-shift methods require a number of circuit evaluations that grows quadratically with the number of trainable parameters, making hardware-based optimisation impractical beyond small system sizes. In this work, we introduce a training framework that reduces this cost to logarithmic in the number of qubits, making gradient-based QNN optimisation feasible on near-term hardware at increasing scales. Our framework combines three co-designed ingredients: (i) a structured, subspace-preserving Butterfly circuit architecture with $O(n \log n)$ parameters and logarithmic depth; (ii) a layer-wise training strategy that confines on-hardware optimisation to one small, well-structured layer at a time; and (iii) a parallelised parameter-shift rule that exploits the commuting structure within each Butterfly layer to extract all gradients in a constant number of circuit executions. Together these reduce the number of distinct circuit evaluations per optimisation step from $O(n^2)$ to $O(\log n)$. We validate the framework on clinical data imputation using the MIMIC-III electronic health record dataset, a demanding benchmark sensitive to optimisation instability and model variance. Hybrid classical-quantum models are trained directly on IonQ Forte Enterprise trapped-ion hardware at 16 qubits without performance degradation relative to ideal or noisy simulation and via tensor-network simulation at 32 qubits, with 32-qubit inference executed on hardware. The resulting models match or exceed strong classical neural baselines in downstream patient survival prediction while exhibiting reduced variance across runs, demonstrating that the proposed framework enables practical, scalable QNN training under realistic hardware constraints.

Parity (XOR) classification requires detecting discrete, high-order feature interactions that smooth classical kernels cannot efficiently capture. We study how quantum kernel advantage depends on parity complexity, the number of features entering the XOR rule, and find a clear threshold behavior. We pair a ZZ quantum feature map with binary {0, pi} encoding (features median thresholded before circuit input) to expose parity structure. A binary encoding ablation, RBF SVM trained on the identical {0, pi} features, separates encoding from circuit effects: at low complexity (n = 5 features), binary RBF achieves 83.4% +/- 1.7% and the quantum kernel 81.2% +/- 1.9%, showing encoding drives performance there. At high complexity (n = 11 features, 11 qubits, r = 3 ZZ repetitions), all classical methods collapse to near-random (approx. 50%), binary RBF reaches only 54.3% +/- 1.1%, and the quantum ZZ kernel achieves 66.3% +/- 3.2% (mean +/- std, 10 seeds), a +12.0 percentage-point margin over the binary ablation and approx. 7x higher kernel-target alignment (0.094 +/- 0.020 vs. 0.013 +/- 0.001). These results identify parity complexity as a concrete axis along which genuine quantum kernel advantage, not attributable to encoding alone, emerges.