Impact of Barren Plateaus on the Hessian and Higher Order Derivatives
Marco Cerezo, Patrick J. Coles · arXiv (Cornell University) · 2020
The Barren Plateau (BP) phenomenon is an issue for certain quantum neural networks and variational quantum algorithms, whereby the gradient vanishes exponentially in the system size $n$. The question of whether high-order derivative information such as the Hessian could help escape a BP was recently posed in the literature. Here we show that the elements of the Hessian are exponentially suppressed in a BP, so estimating the Hessian in this situation would require a precision that scales exponentially with $n$. Hence, Hessian-based approaches do not circumvent the exponential scaling associated with BPs. We also show the exponential suppression of higher order derivatives. Hence, BPs will impact optimization strategies that go beyond (first-order) gradient descent.