Estimates of Loss Function Concentration in Noisy Parametrized Quantum Circuits

Variational quantum computing offers a powerful framework with applications across diverse fields such as quantum chemistry, machine learning, and optimization. However, its scalability is hindered by the exponential concentration of the loss function, known as the barren plateau problem. While significant progress has been made and prior work has separately analyzed barren plateaus in unitary and noisy settings, their combined impact remains poorly understood, largely due to limitations in conventional Lie-algebraic approaches. In this work, we introduce an analytical framework based on non-negative matrix theory that enables the description of the variance in layered noisy quantum circuits with arbitrary noise channels. This approach enables the derivation of exact expressions in the deep-circuit regime, uncovering the complex interplay between unitary layers and noise. Notably, we identify a noise-induced absorption mechanism—a phenomenon absent in purely unitary dynamics—which provides new insight into how noise shapes circuit behavior. We further present a controlled convergence analysis, establishing general lower bounds on the variance of both deep and shallow circuits. This leads to a principled connection between noise resilience and the expressive capacity of parameterized quantum circuits, particularly under smart initialization strategies. Our theoretical results are supported by numerical simulations and illustrative applications.