VARIATIONAL QUANTUM ALGORITHMS ARE LIPSCHITZ SMOOTH

Christopher Kverne, Mayur Akewar, Nicholas S. DiBrita, Yuqian Huo, Tirthak Patel, Janki Bhimani

Keywords: Variational Quantum Algorithms, L-smoothness, Optimization Landscapes

Abstract: The successful gradient-based training of Variational Quantum Algorithms (VQAs) hinges on the $L$-smoothness of their optimization landscapes—a property that bounds curvature and ensures stable convergence. While $L$-smoothness is a common assumption for analyzing VQA optimizers, there has been a need for a more direct proof for general circuits, a tighter bound for practical guidance, and principled methods that connect landscape geometry to circuit design. We address these gaps with Four core contributions. First, we provide an intuitive proof of L-smoothness and derive a new bound on the smoothness constant, $L \le 4||M||_{2}\sum_{k=1}^{P}||G_{k}||_{2}^{2}$, that is never looser and often strictly tighter than previously known. Second, we show that this bound reliably predicts the scaling behavior of curvature in deep circuits and identify a saturation effect that serves as a direct geometric signature of inefficient overparameterization. Third, we leverage this predictable scaling to introduce an efficient heuristic for setting near-optimal learning rates. Fourth we demonstrate that our heuristic remains robust in noisy environments enabling Adam and SGD to achieve convergence rates competitive with the Quantum Natural Gradient optimizer.

Primary Area: optimization

Submission Number: 15126