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=Theta(varepsilon-2), and the peak-qubit requirement qpeak=maxD,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 ≤ 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 4.3× 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.