Oral: Efficient Estimation of Cost Function and its n^th-order Derivatives in Variational Quantum Algorithms

Variational quantum algorithms (VQAs) have garnered significant attention for their potential to address complex problems across diverse fields. However, challenges posed by classical optimization techniques, such as barren plateaus, hinder the effective implementation of VQAs. In this work, we propose an efficient scheme, based on a weighted sum of unitary operators, to evaluate the cost function and its arbitrary derivatives within the framework of VQAs. This approach enables the formulation of a direct optimization routine, which we demonstrate to outperform the non-gradient-based COBYLA method in applications involving dynamics and steady-state problems of nonlinear systems. Specifically, we investigate the nonlinear dynamics of fluid configurations governed by the Burgers' equation and the minimum energy steady-state problem of the nonlinear Schrödinger equation across various parameter regimes. Our findings indicate that the proposed scheme effectively circumvents issues related to barren plateaus and local minima, challenges that the COBYLA optimizer fails to overcome. Our approach is very generic and may advance the practical implementation and progression of variational algorithms on NISQ devices.