Linking barren plateaus to effective parameters in deep rotation-gate-based parameterized quantum circuits

The barren plateaus phenomenon, where the gradients of parametrized quantum circuits become vanishingly small, poses a significant challenge in quantum machine learning. Foundational analyses of barren plateaus often use methods like the Weingarten formula, which, under the global Haar random assumption, effectively links the gradient variance to the Hilbert space dimension. However, many practical circuits, such as Hardware-Efficient Ans"atze, utilize a different, non-Haar random model based on the sampling of parameters for single-qubit rotation gates. This study introduces a direct computational method for the gradient expectation and variance, specifically for parametrized quantum circuits constructed from single-qubit rotation gates. Our analysis reveals a direct, quantitative relationship between the gradient variance, circuit depth, and number of effective parameters. Numerical simulations further confirm the validity of our theoretical results. Our findings highlight the role of effective parameters as a key indicator for gradient trainability and provide a precise framework for analyzing barren plateaus in these specific non-Haar random circuits.