In this work, our prime objective is to study the phenomena of quantum chaos and complexity in the machine learning dynamics of Quantum Neural Network (QNN). A Parameterized Quantum Circuits (PQCs) in the hybrid quantum-classical framework is introduced as a universal function approximator to perform optimization with Stochastic Gradient Descent (SGD). We employ a statistical and differential geometric approach to study the learning theory of QNN. The evolution of parametrized unitary operators is correlated with the trajectory of parameters in the Diffusion metric. We establish the parametrized version of Quantum Complexity and Quantum Chaos in terms of physically relevant quantities, which are not only essential in determining the stability, but also essential in providing a very significant lower bound to the generalization capability of QNN. We explicitly prove that when the system executes limit cycles or oscillations in the phase space, the generalization capability of QNN is maximized. Finally, we have determined the generalization capability bound on the variance of parameters of the QNN in a steady state condition using Cauchy Schwartz Inequality.

翻译：在这项工作中,我们的首要目标是研究量子神经网络(QNN)机器学习动态中量子混乱和复杂现象。混合量子古典框架中的量子电路(PQCs)被引入一个通用功能近似器,以便与Stochatistic 梯子(SGD)进行优化。我们采用统计和差分几何方法研究QNN的学习理论。合成单一操作器的演变与扩散指标参数的轨迹相关。我们从物理相关量的角度建立了量子复杂度和量子共振电路(PQQCs)的平衡版,这不仅对于确定稳定性至关重要,而且对于提供与QNNN的普及能力非常低的制约也至关重要。我们明确证明,当系统在使用阶段空间实施限制周期或振荡时,QNN的通用能力是最大化的。最后,我们确定了QNN的通用能力,以QN的参数在稳定状态下,利用Casy Schtz 状态的参数差异为约束。