  Quantum Neural Networks (QNNs) have been recently proposed as generalizations
of classical neural networks to achieve the quantum speed-up. Despite the
potential to outperform classical models, serious bottlenecks exist for
training QNNs; namely, QNNs with random structures have poor trainability due
to the vanishing gradient with rate exponential to the input qubit number. The
vanishing gradient could seriously influence the applications of large-size
QNNs. In this work, we provide a viable solution with theoretical guarantees.
Specifically, we prove that QNNs with tree tensor and step controlled
architectures have gradients that vanish at most polynomially with the qubit
number. We numerically demonstrate QNNs with tree tensor and step controlled
structures for the application of binary classification. Simulations show
faster convergent rates and better accuracy compared to QNNs with random
structures.
