Variational quantum algorithms (VQAs) with random structures have poor trainability due to the exponentially vanishing gradient as the circuit depth and the qubit number increase. This result leads to a general belief that a deep circuit will not be feasible. In this work, we provide a viable solution to the vanishing gradient problem for deep VQAs with theoretical guarantees. Specifically, we prove that for quantum controlled-layer and quantum residual network (QResNet), architectures, the expectation of the gradient norm can be lower bounded by a value that is independent of the qubit number and the circuit depth. Our results follow from a careful analysis of the gradient behavior on parameter space consisting of rotation angles, as employed in almost all VQAs, instead of relying on impractical 2-design assumptions. We conduct several numerical experiments as verifications, where only our circuits are trainable and converge, while hardware-efficient and random circuits with similar number of parameters in comparison cannot converge.