Long Range Frequency Tuning for QML
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866910223095037952 |
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| author | Poppel, Michael Baumann, Markus Wölckert, Sebastian Linnhoff-Popien, Claudia Stein, Jonas |
| author_facet | Poppel, Michael Baumann, Markus Wölckert, Sebastian Linnhoff-Popien, Claudia Stein, Jonas |
| contents | Angle-encoded variational quantum circuits admit a truncated Fourier series representation of their output, but approximating functions with maximum frequency $ω_{\max}$ using fixed unary encoding requires $\mathcal{O}(ω_{\max})$ 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 \emph{ternary grid initialization} -- setting prefactors to $\{1, 3, 9, \ldots, 3^{k-1}\}$ -- which resolves this limitation by ensuring every target frequency within $[-ω_{\max}, ω_{\max}]$ lies within $\tfrac{1}{2}$ 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 $20\times$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_23409 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Long Range Frequency Tuning for QML Poppel, Michael Baumann, Markus Wölckert, Sebastian Linnhoff-Popien, Claudia Stein, Jonas Machine Learning Artificial Intelligence Emerging Technologies Quantum Physics Angle-encoded variational quantum circuits admit a truncated Fourier series representation of their output, but approximating functions with maximum frequency $ω_{\max}$ using fixed unary encoding requires $\mathcal{O}(ω_{\max})$ 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 \emph{ternary grid initialization} -- setting prefactors to $\{1, 3, 9, \ldots, 3^{k-1}\}$ -- which resolves this limitation by ensuring every target frequency within $[-ω_{\max}, ω_{\max}]$ lies within $\tfrac{1}{2}$ 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 $20\times$ 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. |
| title | Long Range Frequency Tuning for QML |
| topic | Machine Learning Artificial Intelligence Emerging Technologies Quantum Physics |
| url | https://arxiv.org/abs/2602.23409 |