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

Qingyu Meng, Yangshuai Wang

TL;DR

The paper tackles trainability bottlenecks in quantum regression by introducing a hybrid quantum-classical model with a learnable geometric preconditioner and a curriculum-based optimization that grows circuit depth while transitioning from SPSA to Adam. The approach leverages a lightweight classical embedding to reshape inputs before a data re-uploading quantum feature map, coupled with layer-wise growth and a two-stage optimizer to mitigate barren plateaus and gradient noise. Empirical results on PDE-informed regression and small-data benchmarks show improved convergence stability and reduced structured errors, indicating practical gains under data scarcity and finite-shot regimes. The work underscores the value of joint representation conditioning and optimization schedules for stabilizing quantum regression in near-term settings, with avenues for hardware validation and theoretical analysis of optimization geometry.

Abstract

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.

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

TL;DR

The paper tackles trainability bottlenecks in quantum regression by introducing a hybrid quantum-classical model with a learnable geometric preconditioner and a curriculum-based optimization that grows circuit depth while transitioning from SPSA to Adam. The approach leverages a lightweight classical embedding to reshape inputs before a data re-uploading quantum feature map, coupled with layer-wise growth and a two-stage optimizer to mitigate barren plateaus and gradient noise. Empirical results on PDE-informed regression and small-data benchmarks show improved convergence stability and reduced structured errors, indicating practical gains under data scarcity and finite-shot regimes. The work underscores the value of joint representation conditioning and optimization schedules for stabilizing quantum regression in near-term settings, with avenues for hardware validation and theoretical analysis of optimization geometry.

Abstract

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.
Paper Structure (23 sections, 13 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 23 sections, 13 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Related Works
  3. Methodology
  4. Hybrid Architecture with Classical Embedding
  5. Trainable Quantum Feature Map via Data Re-uploading
  6. Curriculum-Driven Optimization Protocol
  7. Training objective.
  8. Curriculum optimization.
  9. Stage 1: Stochastic approximation via SPSA.
  10. Stage 2: Analytic fine-tuning with identity initialization.
  11. Layer-wise growth strategy.
  12. Numerical Experiments
  13. Experimental Setup
  14. Scientific Tasks.
  15. Efficiency Metrics
  16. ...and 8 more sections

Figures (3)

  • Figure 1: Hybrid Quantum Regression Framework. A lightweight classical embedding transforms inputs before quantum encoding to improve conditioning for the variational circuit. Training follows a curriculum that gradually increases circuit depth and transitions from SPSA-based exploration to Adam-based refinement.
  • Figure 2: Numerical validation on PDE benchmarks. We show ground truth and absolute error distributions for four benchmarks: (A) 2D Poisson, (B) 2D Convection--Diffusion, (C) 2D Nonlinear Poisson, and (D) 3D Modified Helmholtz. Under the fixed training protocol, the Hybrid QNN yields lower and less structured error patterns compared to the considered baselines.
  • Figure 3: Optimization dynamics on Yacht (representative fold). Training loss trajectories for SPSA-only, Adam-only, and the proposed two-stage schedule.