Physics-Constrained Quantum Circuit Born Machines for Synthetic Industrial Sensor Data Under Limited Training Samples: A Controlled Comparison With Classical Generators

Industrial condition-monitoring programmes rarely possess the volume of labelled fault data that modern generative models require, and the few records of rare faults that do exist are often too sensitive to share. This paper develops and evaluates a physics-constrained quantum generative framework for producing synthetic bearing-condition records from very small training sets. A fourteen-qubit conditional quantum circuit Born machine (QCBM) learns the joint distribution of a fault label and four Gray-code-embedded vibration condition indicators (log-RMS, kurtosis, and the envelope-spectrum energy ratios at the outer-race and inner-race characteristic frequencies). Physical admissibility is enforced through a feasibility support derived from a parameter-envelope sweep of a rolling-element bearing impulse-response model, applied both as a soft penalty inside a maximum-mean-discrepancy objective and as a measurement post-selection at inference. The generator is trained with exact adjoint gradients and, in a hardware-realistic variant, with parameter-shift gradients estimated from 500 to 2000 measurement shots, and is sampled under depolarising gate noise. On training sets of 32 to 256 records with a 5% rare inner-race class, the framework is benchmarked against Gaussian-mixture, kernel-density, conditional-GAN and a parameter-matched classical energy model trained with the identical objective, using fidelity, diversity, physical validity, membership-inference privacy and train-on-synthetic/test-on-real utility. The projected QCBM eliminates physically impossible records, raises downstream macro-F1 by 0.24 on average over the unprojected circuit, lifts rare-class recall to the level of the best classical augmentation, and outperforms the parameter-matched classical energy model on fidelity at 32 to 64 records and on privacy at every size; it does not, however, surpass Gaussian-mixture or kernel-density generators on fidelity, coverage or utility on this four-indicator space, and all results are obtained by classical simulation of the circuit. The study therefore delimits where quantum generative models presently help, and identifies the feasibility constraint as the generator-agnostic ingredient that matters most under scarce data.