Geometric Preconditioning and Curriculum Optimization for Trainable Variational Quantum Regression

TL;DR

This paper introduces a hybrid quantum-classical approach with geometric preconditioning and curriculum optimization to improve the trainability of variational quantum regression models, demonstrating lower errors on benchmarks.

Contribution

It proposes a learnable geometric preconditioner combined with a curriculum protocol to enhance quantum regression training, supported by theoretical and empirical analysis.

Findings

Hybrid QNN reduces error compared to pure QNNs under similar budgets.

Classical methods remain competitive and sometimes outperform the hybrid quantum approach.

The approach improves trainability by reshaping input distributions and progressively increasing circuit depth.

Abstract

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.