Quantum-data-driven dynamical transition in quantum learning

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 classes. 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.