qc-kmeans: A Quantum Compressive K-Means Algorithm for NISQ Devices

Clustering on NISQ hardware is constrained by data loading and limited qubits. We present qc-kmeans, a hybrid compressive k-means that summarizes a dataset with a constant-size Fourier-feature sketch and selects centroids by solving small per-group QUBOs with shallow QAOA circuits. The QFF sketch estimator is unbiased with mean-squared error O\(varepsilon²\) for B,S=Θ\(varepsilon^-2\), and the peak-qubit requirement q_peak=max\{D,lceil log₂ Brceil + 1\} does not scale with the number of samples. A refinement step with elitist retention ensures non-increasing surrogate cost. In Qiskit Aer simulations depth $p{=}1$, the method ran with le 9 qubits on low-dimensional synthetic benchmarks and achieved competitive sum-of-squared errors relative to quantum baselines; runtimes are not directly comparable. On nine real datasets up to $4.3times 10⁵$ points, the pipeline maintained constant peak-qubit usage in simulation. Under IBM noise models, accuracy was similar to the idealized setting. Overall, qc-kmeans offers a NISQ-oriented formulation with shallow, bounded-width circuits and competitive clustering quality in simulation.