Establish a rigorous theory for neural network–generated parameters mitigating barren plateaus in VQAs

Establish a rigorous mathematical and computational theory explaining why classical neural networks used to generate parameters for parameterized quantum circuits mitigate barren plateaus in variational quantum algorithms, including the mechanism by which neural network–generated parameters enable smoother optimization trajectories and avoid vanishing gradients as system size increases.

Background

Variational quantum algorithms often suffer from barren plateaus, where gradients vanish exponentially with the number of qubits, hampering trainability. A promising mitigation strategy is to use classical neural networks to generate parameters for parameterized quantum circuits, which has empirically shown faster convergence and improved trainability compared to random initialization.

While these empirical successes are documented, a clear theoretical foundation articulating why and how neural network–generated parameters alleviate barren plateaus has not been fully established. This paper proposes a Lie-theoretic geometric perspective on single-qubit gates as elements of SU(2), aiming to provide a mathematical explanation, but explicitly notes that the overarching theory for the approach remains unclear, motivating the formulation of this open problem.