量子数据决定量子神经网络训练能否收敛
Quantum-data-driven dynamical transition in quantum learning
- 通过分析动力学方程固定点，发现量子数据影响训练路径
- 揭示七种不同动态行为，存在指数与多项式收敛两类
- 适合研究量子优化和量子机器学习的学者参考
参数化量子电路在特定代价函数下优化的量子神经网络（QNN）为实现近中期量子优势提供了新范式。理解其训练动力学对优化性能至关重要，但量子数据在监督学习（如分类与回归）中的作用仍不明确。本文揭示了一种由量子数据驱动的动力学相变现象，目标值与数据共同决定训练是否收敛。通过对动力学方程固定点的解析分类，构建了包含七种不同动态行为的完整‘相图’，源于多余维数的分岔。微扰分析识别出指数与多项式两类收敛模式。我们提出非微扰理论，通过广义受限哈亚尔随机矩阵族解释该相变。分析结果经数值模拟及在IBM量子设备上的实验验证。研究成果为设计加速收敛的代价函数提供指导。
原文摘要 · Abstract (English)
Quantum neural networks, parameterized quantum circuits optimized under a specific cost function, provide a paradigm for achieving near-term quantum advantage in quantum information processing. Understanding QNN training dynamics is crucial for optimizing their performance, however, the role of quantum data in training for supervised learning such as classification and regression remains unclear. We reveal a quantum-data-driven dynamical transition where the target values and data determine the convergence of the training. Through analytical classification over the fixed points of the dynamical equation, we reveal a comprehensive `phase diagram' featuring seven distinct dynamics originating from a bifurcation with multiple codimension. Perturbative analyses identify both exponential and polynomial convergence class. We provide a non-perturbative theory to explain the transition via generalized restricted Haar ensemble. The analytical results are confirmed with numerical simulations and experimentation on IBM quantum devices. Our findings provide guidance on constructing the cost function to accelerate convergence in QNN training.
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