  Variational quantum algorithms rely on gradient based optimization to
iteratively minimize a cost function evaluated by measuring output(s) of a
quantum processor. A barren plateau is the phenomenon of exponentially
vanishing gradients in sufficiently expressive parametrized quantum circuits.
It has been established that the onset of a barren plateau regime depends on
the cost function, although the particular behavior has been demonstrated only
for certain classes of cost functions. Here we derive a lower bound on the
variance of the gradient, which depends mainly on the width of the circuit
causal cone of each term in the Pauli decomposition of the cost function. Our
result further clarifies the conditions under which barren plateaus can occur.
