Quantum kernel support vector machines for trabecular bone classification: comparing feature reduction strategies on synthetic micro-CT data

We compare five dimensionality reduction strategies as preprocessing for quantum kernel SVMs: principal component analysis (PCA), Gaussian random projection (RP Gaussian), sparse random projection (RP Sparse), partial least squares (PLS), and uniform manifold approximation and projection (UMAP). These span four methodological families: linear variance-maximising, random distance-preserving, supervised covariance-maximising, and nonlinear manifold-preserving.

Input data consists of 43-dimensional texture feature vectors extracted from synthetic trabecular bone micro-CT slices, comprising 31 first-order statistical descriptors and 12 grey-level co-occurrence matrix (GLCM) features. Each reduction method compresses these to 8 dimensions (4 for PLS, limited by target count). Reduced features are standardised via RobustScaler and rescaled to [0, π] via min-max scaling to match the periodic encoding of the quantum circuit.

The quantum register is prepared using a ZZFeatureMap with 2 repetitions and linear entanglement on 8 qubits (4 for PLS). Each data point is encoded by applying single-qubit rotations parameterised by the reduced feature values, followed by entangling ZZ interaction gates that create pairwise correlations between qubits. The quantum kernel is computed as the fidelity (squared overlap) between pairs of quantum states prepared from different data points, yielding a full 500×500 kernel matrix per reduction method via Qiskit's FidelityQuantumKernel on a statevector simulator.

Classification is performed using precomputed quantum kernel matrices with a support vector machine (SVM) optimiser for two binary tasks: BV/TV classification (sparse versus dense bone) and trabecular number classification (low versus high). Classical RBF-kernel SVMs serve as baselines, evaluated on identical fold partitions using 5×5 repeated stratified cross-validation with matched random seeds. Statistical comparison employs three tests accounting for fold dependence: uncorrected Wilcoxon signed-rank, Nadeau–Bengio corrected paired t-test, and Dietterich 5×2 CV paired t-test.

To address the limitations of repeated cross-validation on a single dataset, we additionally perform independent dataset validation: 10 fully independent datasets of 500 samples each are generated with separate random seeds, and the entire pipeline - feature extraction, reduction fitting, kernel computation, and SVM evaluation - is executed independently for each. Beyond classification, quantum kernel ridge regression is evaluated for continuous BV/TV prediction to test whether the ZZ kernel captures smooth metric structure.

We demonstrate the correct functioning of our quantum-assisted approach by implementing and running the quantum kernel algorithm on a statevector quantum simulator via Qiskit's FidelityQuantumKernel. The statevector simulator computes exact quantum state fidelities without shot noise, providing the theoretical best-case performance of the quantum kernel and isolating the effect of dimensionality reduction from hardware-specific artefacts.

For each of the four reduction methods carried through to full evaluation (PCA, RP Gaussian, PLS, UMAP), a complete 500×500 quantum kernel matrix was computed, requiring approximately 140 minutes per matrix (27.7 hours total for the 10 independent validation datasets). The quantum SVM was then evaluated using 5×5 repeated stratified cross-validation (25 folds) on identical partitions as the classical baseline, ensuring a fair paired comparison.

The results demonstrate that the quantum pipeline functions correctly across all reduction methods, with quantum SVM accuracies ranging from 0.620 to 0.725 for BV/TV classification and 0.921 to 0.976 for Tb.N classification - all substantially above the 0.50 random baseline. Kernel matrix diagnostics confirm that the quantum circuits produce physically meaningful kernel matrices with properties that vary systematically across reduction methods: effective rank ratios range from 0.179 (PLS) to 0.831 (PCA), and off-diagonal means range from 0.010 to 0.089, consistent with theoretical predictions about exponential concentration in quantum kernels.

The key finding is that UMAP is the only reduction method where the quantum kernel remains competitive with the classical RBF-SVM baseline. Under repeated cross-validation, UMAP showed a +0.032 accuracy gap favouring the quantum kernel (Dietterich 5×2 CV p = 0.177), while all linear methods showed substantial quantum deficits of −0.090 to −0.116. Independent validation on 10 separately generated datasets confirmed that UMAP maintains quantum–classical parity (gap = −0.030, p = 0.123), whereas the linear method deficits remained significant even under corrected statistical tests. Quantum kernel ridge regression was additionally evaluated, revealing uniform failure at regression (negative R² for all 8-qubit methods), demonstrating that the ZZ kernel captures decision boundaries but not smooth metric structure.

All code, including the synthetic bone generator, feature extraction, reduction pipeline, quantum kernel computation, and statistical analysis, is publicly available on GitHub and archived on Zenodo under the MIT licence, enabling full reproducibility.

Our approach establishes a practical hybridisation strategy combining classical preprocessing with quantum kernel computation. The pipeline is inherently hybrid: classical computers handle feature extraction (43 texture descriptors from micro-CT slices), dimensionality reduction (UMAP compression to 8 dimensions), and SVM optimisation, while the quantum processor computes kernel matrix entries via state fidelity estimation. This division assigns each component to the hardware best suited for it and is directly compatible with near-term quantum devices.

Scaling to realistic problem sizes is feasible along several axes. The current 8-qubit ZZFeatureMap can be executed on existing quantum hardware (devices with 20+ qubits are commercially available), and the kernel matrix computation is embarrassingly parallel - each of the O(n²) matrix entries is an independent circuit execution - enabling efficient distribution across multiple quantum processing units. For a dataset of n samples, the quantum bottleneck is n(n+1)/2 circuit evaluations, each requiring depth proportional to the number of qubits and feature map repetitions. At 8 qubits with 2 repetitions, circuit depth remains well within the coherence limits of current superconducting and trapped-ion platforms.

However, our results also reveal important limitations that temper near-term optimism. The quantum kernel did not achieve a statistically significant advantage over the classical RBF-SVM for any reduction method - UMAP achieved parity, not superiority. This is consistent with theoretical analyses showing that quantum kernel advantages require structural alignment between data geometry and the quantum feature space, which cannot be guaranteed for arbitrary datasets. Furthermore, the uniform failure at regression (negative R² for all 8-qubit methods) demonstrates that exponential concentration of kernel values poses a fundamental barrier to continuous prediction tasks at moderate qubit counts, as the kernel matrix approaches the identity and loses the smooth metric structure regression requires.

These findings have direct implications for scaling. Increasing qubit count to handle higher-dimensional features will exacerbate exponential concentration unless accompanied by kernel designs less susceptible to this phenomenon - for instance, data-dependent variational encodings or amplitude-based feature maps. The critical insight from our work is that the preprocessing strategy (specifically, nonlinear manifold-preserving reduction such as UMAP) can mitigate concentration effects by pre-structuring the input data into clusters that the quantum kernel can distinguish, effectively shifting part of the computational burden to the classical preprocessing stage. This suggests that practical quantum advantage in kernel methods will likely require co-optimisation of the reduction method, encoding circuit, and downstream task rather than improvements to the quantum component alone.