Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus

Parametrized quantum circuits serve as ansatze for solving variational problems and provide a flexible paradigm for the programming of near-term quantum computers. Ideally, such ansatze should be highly expressive, so that a close approximation of the desired solution can be accessed. On the other hand, the ansatz must also have sufficiently large gradients to allow for training. Here, we derive a fundamental relationship between these two essential properties: expressibility and trainability. This is done by extending the well-established barren plateau phenomenon, which holds for ansatze that form exact 2-designs, to arbitrary ansatze. Specifically, we calculate the variance in the cost gradient in terms of the expressibility of the ansatz, as measured by its distance from being a 2-design. Our resulting bounds indicate that highly expressive ansatze exhibit flatter cost landscapes and therefore will be harder to train. Furthermore, we provide numerics illustrating the effect of expressibility on gradient scalings and we discuss the implications for designing strategies to avoid barren plateaus.