Toward Joint Optimization of Circuit Depth and Training Data Size in Adaptively Grown Quantum Classifiers

Building a quantum model involves a tradeoff: how complex the circuit should be, and how much training data it needs. Caro et al. show that models with fewer trainable gates need less training data to generalize well. Q-FLAIR shows that a quantum feature-map circuit can be grown gate-by-gate, stopping once further growth stops improving the training loss. We ask whether these two results combine into a predictable scaling law. Does Q-FLAIR's own stopping rule pick larger or smaller circuits as training data grows? Does the resulting generalization behavior track Caro et al.'s bound? We reimplement Q-FLAIR's growth mechanism faithfully, including its analytic reconstruction and exact stopping rule. We run it on full-resolution (784-pixel) MNIST 3-vs-5 classification, at five training-set sizes from N=2000 to 10000. We then fine-tune each resulting circuit, so we can measure Caro et al.'s notion of active gates, K. We find no predictable relationship between training-set size and the circuit size Q-FLAIR converges to. Circuit size and test accuracy both vary non-monotonically with N, and seed-to-seed variance is nearly as large as any trend across N. The empirical generalization gap never exceeds Caro et al.'s bound in 14 of 15 runs, so the bound holds as a valid guarantee. But the gap correlates only weakly with the bound's value (r=0.12). This shows that K does not explain most of the variation we observe. Why a valid guarantee can coexist with such weak predictive power remains an open question, and answering it may be necessary before circuit depth and training data size can be jointly optimized in practice.