How Far Can a Trainable Frequency Move? Resolution, Grids and Coupling in Trainable-FrequencyQuantum Models

arXiv:2602.23409v4 Announce Type: replace-cross 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 coeffici

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