可训练频率量子模型能走多远：分辨率、网格与耦合

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Abstract:Trainable-frequency quantum models train the prefactors of their encoding gates so that the circuit's frequencies can move onto the target's. From the unit start of prior work, training rarely reaches a target more than one resolution step away. We show that the target-driven part of the prefactor gradient is a weighted sum of slopes of the overlap $\sin u/u$, where $u$ is the model-target frequency distance scaled by the window half-width. This gives each frequency a reach: under gradient flow, an isolated real frequency whose coefficient adapts optimally reaches its target from within half a step, for every target at least one step from zero frequency; with a fixed coefficient, a phase-dependent barrier can stop it. A ternary prefactor grid places a model frequency within half a step of every band frequency, with logarithmically many encoding gates. Single-frequency circuits trained with Adam and jointly trained amplitudes behave as predicted: inside the first side lobe, a frequency starts toward its target exactly when in phase with it in 98% of runs, and only 11 of 768 frequencies starting beyond the lobe reached their target. With several frequencies, the grid start raised successful fits on synthetic targets from 69% to 84% against the best random start. With many target frequencies, least squares on the fixed grid fits near-perfectly, but the circuit does not reach the coefficients; training the prefactors narrows this shortfall, but with 8 and 24 target frequencies, training the fixed grid ten times longer fitted better. On real series, training the frequencies improved held-out $R^2$ at equal budget by 0.010--0.045 on four of seven series, but a classical screen did not separate these from the others. A classical model with independently trainable frequencies shows the same dependence on the start; our results describe how to train such models, not an advantage of the circuit.

Comments:	v4: substantially revised; new title (previously: Long Range Frequency Tuning for QML); new theory with Lean formalisation, new experiments and real-data protocol. Code: doi:https://doi.org/10.5281/zenodo.23974655
Subjects:	Machine Learning (cs.LG); Artificial Intelligence (cs.AI); Emerging Technologies (cs.ET); Quantum Physics (quant-ph)
Cite as:	arXiv:2602.23409 [cs.LG]
	(or arXiv:2602.23409v4 [cs.LG] for this version)
	https://doi.org/10.48550/arXiv.2602.23409 arXiv-issued DOI via DataCite

Submission history

From: Michael Poppel [view email]
[v1] Thu, 26 Feb 2026 14:56:37 UTC (2,509 KB)
[v2] Fri, 15 May 2026 07:35:47 UTC (3,151 KB)
[v3] Mon, 20 Jul 2026 09:19:14 UTC (3,205 KB)
[v4] Thu, 5 Nov 2026 10:50:42 UTC (1,395 KB)