  Parameterized quantum circuits serve as ans\"{a}tze for solving variational
problems and provide a flexible paradigm for programming near-term quantum
computers. Ideally, such ans\"{a}tze 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 ans\"{a}tze that form
exact 2-designs, to arbitrary ans\"{a}tze. 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 ans\"{a}tze exhibit flatter cost landscapes and
therefore will be harder to train. Furthermore, we provide numerics
illustrating the effect of expressiblity on gradient scalings, and we discuss
the implications for designing strategies to avoid barren plateaus.
