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Explore the theoretical foundations and practical applications of Fourier fingerprints in quantum machine learning through this 21-minute conference presentation from QTML 2025. Delve into how parameterized quantum circuits (PQCs) function as machine learning models, examining the critical relationship between quantum feature maps that encode classical inputs into Hilbert space and variational ansatzes that manipulate these inputs through trainable parameterized gates. Understand how quantum Fourier models (QFMs) represent their outputs as partial Fourier series and discover why the choice of feature map and ansatz significantly impacts the Fourier spectrum's performance in frameworks like quantum neural networks and quantum kernels. Learn about the theoretical motivation behind Fourier coefficient correlations in QFMs and their implications for independent control of Fourier series terms. Examine the computation of these correlations across popular ansatzes and discover how each ansatz produces a unique pattern called the Fourier fingerprint. Investigate the practical utility of Fourier coefficient correlation (FCC) derived from these fingerprints for learning random 1D and 2D Fourier series, with demonstrations showing how FCC effectively predicts optimal ansatz performance. Apply these concepts to the challenging problem of 2D jet reconstruction in high-energy physics, where FCC proves capable of predicting the best performing ansatz before training begins, establishing Fourier fingerprints as a powerful tool for optimal ansatz selection in quantum machine learning.
