  In variational quantum algorithms the parameters of a parameterized quantum
circuit are optimized in order to minimize a cost function that encodes the
solution of the problem. The barren plateau phenomenon manifests as an
exponentially vanishing dependence of the cost function with respect to the
variational parameters, and thus hampers the optimization process. We discuss
how, and in which sense, the phenomenon of noise-induced barren plateaus
emerges in parameterized quantum circuits with a layered noise model. Previous
results have shown the existence of noise-induced barren plateaus in the
presence of local Pauli noise [arXiv:2007.14384]. We extend these results
analytically to arbitrary completely-positive trace preserving maps in two
cases: 1) when a parameter-shift rule holds, 2) when the parameterized quantum
circuit at each layer forms a unitary $2$-design. The second example shows how
highly expressive unitaries give rise not only to standard barren plateaus
[arXiv:1803.11173], but also to noise-induced ones. In the second part of the
paper, we study numerically the emergence of noise-induced barren plateaus in
QAOA circuits focusing on the case of MaxCut problems on $d$-regular graphs and
amplitude damping noise.
