Connecting Ansatz Expressibility to Gradient Magnitudes and Barren Plateaus

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