Avoiding Barren Plateaus Using Classical Shadows

Abstract

Variational quantum algorithms are promising algorithms for achieving quantum advantage on near-term devices. The quantum hardware is used to implement a variational wave function and measure observables, whereas the classical computer is used to store and update the variational parameters. The optimization landscape of expressive variational ansätze is however dominated by large regions in parameter space, known as barren plateaus, with vanishing gradients, which prevents efficient optimization. In this work we propose a general algorithm to avoid barren plateaus in the initialization and throughout the optimization. To this end we define a notion of weak barren plateaus (WBPs) based on the entropies of local reduced density matrices. The presence of WBPs can be efficiently quantified using recently introduced shadow tomography of the quantum state with a classical computer. We demonstrate that avoidance of WBPs suffices to ensure sizable gradients in the initialization. In addition, we demonstrate that decreasing the gradient step size, guided by the entropies allows WBPs to be avoided during the optimization process. This paves the way for efficient barren plateau-free optimization on near-term devices. (LW). LW encounters a WBP for both learning rates that we consider (green star). In contrast, SA avoids the WBP for both learning rates. Good performance and further convergence in the local minimum is only achieved through a smaller learning rate of η = 0.01. We use a system size of N = 10 and a random circuit from Eq. (1) with circuit depth p = 100. Data is averaged over 100 random instances.