Geometric Preconditioning and Curriculum Optimization for Trainable Variational Quantum Regression

Variational quantum circuits are increasingly studied as continuous-function approximators, but quantum regression remains difficult to train when global losses, finite-shot stochasticity, and circuit-depth growth combine to produce weak or ill-conditioned gradient signals. We study this trainability problem in a controlled hybrid quantum--classical regression design. The central ingredient is a capacity-controlled classical embedding that acts as a learnable geometric preconditioner: it reshapes the input distribution seen by a data-reuploading variational circuit while preserving a low-dimensional quantum bottleneck. We pair this representation design with a curriculum protocol that grows circuit depth progressively and switches from SPSA-based stochastic exploration to Adam-based analytic-gradient fine-tuning. We formalize the mechanism through a local quantum-tangent contraction statement: in the linearized quantum-parameter dynamics, the embedding changes the empirical Gram matrix that controls residual contraction and one-step loss decrease. Across finite-size statevector audits on PDE-informed regression benchmarks and small-data tabular tasks, the Hybrid QNN lowers error relative to Pure QNN baselines under matched quantum-model budgets. Strong classical references remain competitive, and in several cases are better in absolute error; the evidence therefore supports a trainability claim for the hybrid QNN design rather than a claim of classical or hardware quantum advantage.