Long Range Frequency Tuning for QML

arXiv:2602.23409v2 Announce Type: replace-cross Abstract: Angle-encoded variational quantum circuits admit a truncated Fourier series representation of their output, but approximating functions with maximum frequency omega_{max} using fixed unary encoding requires O(omega_{max}) encoding gates. Trainable-frequency (TF) circuits promise a reduction by learning the data-encoding prefactors alongside the ansatz parameters, adapting the accessible frequency spectrum to the target during training. We identify a practical barrier that prevents this promise from being realized: the prefactor gradient is suppressed by the spectral gap between the circuit's accessible frequencies and the target spectrum, independently of the ansatz parameters, confining gradient-driven prefactor movement to a narrow neighborhood of initialization. We propose ternary grid initialization -- setting prefactors to {1, 3, 9, ldots, 3^{k-1}} -- which resolves this limitation by ensuring every target frequency within [-omega_{max}, omega_{max}] lies within frac{1}{2} unit of a grid point at initialization, removing the spectral gap suppression by construction. On a synthetic benchmark with target frequencies shifted well beyond the standard initialization range, ternary initialization achieves median R^2 = 0.997 versus 0.18 for unary initialization, with 100% of runs achieving R^2 > 0.95 against 0%. CMA-ES with 20imes the evaluation budget reaches only 25% success, confirming the limitation is a property of the optimization landscape rather than of gradient-based optimization specifically. Real-world validation on two benchmark datasets demonstrates consistent advantages over both fixed and trainable unary baselines.

Source: arXiv cs.AI | 2026-05-18