  Several architectures have been proposed for quantum neural networks (QNNs),
with the goal of efficiently performing machine learning tasks on quantum data.
Rigorous scaling results are urgently needed for specific QNN constructions to
understand which, if any, will be trainable at a large scale. Here, we analyze
the gradient scaling (and hence the trainability) for a recently proposed
architecture that we called dissipative QNNs (DQNNs), where the input qubits of
each layer are discarded at the layer's output. We find that DQNNs can exhibit
barren plateaus, i.e., gradients that vanish exponentially in the number of
qubits. Moreover, we provide quantitative bounds on the scaling of the gradient
for DQNNs under different conditions, such as different cost functions and
circuit depths, and show that trainability is not always guaranteed.
