Trainability-Oriented Hybrid Quantum Regression via Geometric Preconditioning and Curriculum Optimization

Quantum neural networks (QNNs) have attracted growing interest for scientific machine learning, yet in regression settings they often suffer from limited trainability under noisy gradients and ill-conditioned optimization. We propose a hybrid quantum-classical regression framework designed to mitigate these bottlenecks. Our model prepends a lightweight classical embedding that acts as a learnable geometric preconditioner, reshaping the input representation to better condition a downstream variational quantum circuit. Building on this architecture, we introduce a curriculum optimization protocol that progressively increases circuit depth and transitions from SPSA-based stochastic exploration to Adam-based gradient fine-tuning. We evaluate the approach on PDE-informed regression benchmarks and standard regression datasets under a fixed training budget in a simulator setting. Empirically, the proposed framework consistently improves over pure QNN baselines and yields more stable convergence in data-limited regimes. We further observe reduced structured errors that are visually correlated with oscillatory components on several scientific benchmarks, suggesting that geometric preconditioning combined with curriculum training is a practical approach for stabilizing quantum regression.

💡 Research Summary

This paper addresses two major challenges that quantum neural networks (QNNs) face in regression tasks, especially in scientific machine learning (SciML): noisy gradient estimates and barren‑plateau‑induced ill‑conditioning. The authors propose a hybrid quantum‑classical regression framework that combines a lightweight classical embedding with a variational quantum circuit, and they introduce a curriculum‑driven optimization protocol that gradually increases circuit depth while switching from a stochastic optimizer (SPSA) to a gradient‑based optimizer (Adam).

The classical component is a small multi‑layer perceptron (MLP) that maps the high‑dimensional input x∈ℝ^d to a low‑dimensional latent vector z∈ℝ^p, where p is chosen to match the number of qubits required for encoding (p = n_q). This MLP is deliberately capacity‑limited (few layers, few parameters) and is interpreted as a learnable geometric preconditioner: by reshaping the geometry of the data before quantum encoding, it improves the conditioning of the downstream loss landscape with respect to the quantum parameters. The quantum part consists of a data‑re‑uploading variational circuit built from L layers; each layer applies input‑dependent single‑qubit rotations (parameterized by trainable scaling factors ϕ and offsets β) followed by a fixed entangling block (e.g., a linear CNOT topology). The circuit outputs a scalar observable y_q, which is combined with the latent vector z via a linear readout ˆy = wᵀ