The effect of the number of parameters and the number of local feature patches on loss landscapes in distributed quantum neural networks

Quantum neural networks hold promise for tackling computationally challenging tasks that are intractable for classical computers. However, their practical application is hindered by significant optimization challenges, arising from complex loss landscapes characterized by barren plateaus and numerous local minima. These problems become more severe as the number of parameters or qubits increases, hampering effective training. To mitigate these optimization challenges, particularly for classical data, we distribute overlapping local patches across multiple quantum neural networks, processing each patch with an independent quantum neural network, and aggregating their outputs for prediction. In this study, we investigate how the number of parameters and patches affects the loss landscape geometry of this distributed quantum neural network architecture via theoretical and empirical Hessian analyses and loss landscape visualization. Our results confirm that increasing the number of parameters tends to lead to deeper and sharper loss landscapes. Crucially, we theoretically derive and empirically demonstrate that increasing the number of patches significantly reduces the largest Hessian eigenvalue at minima. Furthermore, our analysis of the full Hessian eigenspectrum reveals a structure consisting of a bulk of near-zero eigenvalues and distinct outlier spikes corresponding to the number of classes, similar to classical deep learning models. These findings suggest that our distributed patch approach acts as a form of implicit structural regularization, promoting optimization stability and potentially enhancing generalization. Our study provides valuable insights into optimization challenges and highlights that the distributed patch approach is a promising strategy for developing more trainable and scalable quantum machine learning models for classical data tasks.