Direct Gradient Computation for Barren Plateaus in Parameterized Quantum Circuits

The barren plateau phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. While previous studies attempted to explain the barren plateau phenomenon using the Weingarten formula, the reliance on the Weingarten formula leads to inaccurate conclusions. In this study, we consider a unitary operator $U$ consisting of rotation gates and perform an exact calculation of the expectation required for the gradient computation. Our approach allows us to obtain the gradient expectation and variance directly. Our analysis reveals that gradient expectations are not zero, as opposed to the results derived using the Weingarten formula, but depend on the number of qubits in the system. Furthermore, we demonstrate how the number of effective parameters, circuit depth, and gradient variance are interconnected in deep parameterized quantum circuits. Numerical simulations further confirm the validity of our theoretical results. Our approach provides a more accurate framework for analyzing quantum circuit optimization.