Quantum Fourier Generative Models Reach Training Scale of 1000+ Qubits via Classical Monte Carlo Log-Likelihood

The paper introduces a train-on-classical, deploy-on-quantum pipeline that replaces MMD objectives with log-likelihood loss estimated through Monte Carlo sampling of Fourier coefficients. This permits scaling beyond the low-frequency moment matching that limited prior IQP and variational approaches. Validation on univariate and bivariate targets reached low total variation distance while classical normalizing flows and diffusion models oversmoothed multimodal structure. Deployment on superconducting processors confirmed functional sampling circuits extracted via inverse quantum Fourier transform. The advance directly addresses NISQ-era training bottlenecks by keeping all optimization steps classical and hardware-efficient. Earlier quantum generative model literature, including works on quantum Boltzmann machines and MMD-trained circuits, repeatedly encountered exponential gradient variance or required prohibitive qubit overhead for loss evaluation. By grounding the model in Fourier analysis and Parseval identity, the method achieves unbiased gradient estimates whose cost grows only linearly with feature dimension. Connecting hardware progress to concrete ML workloads rather than supremacy benchmarks, the approach shows that quantum advantage in generative tasks may first appear through fast, high-fidelity sampling after classical training. Remaining limitations include restriction to continuous variables amenable to Fourier embedding and lack of demonstrated multivariate scaling beyond two dimensions. Next steps require extending the framework to trivariate or higher distributions and benchmarking against state-of-the-art classical autoregressive models on identical hardware deployment metrics.