  Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on
Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is
whether noise on NISQ devices places fundamental limitations on VQA
performance. We rigorously prove a serious limitation for noisy VQAs, in that
the noise causes the training landscape to have a barren plateau (i.e.,
vanishing gradient). Specifically, for the local Pauli noise considered, we
prove that the gradient vanishes exponentially in the number of qubits $n$ if
the depth of the ansatz grows linearly with $n$. These noise-induced barren
plateaus (NIBPs) are conceptually different from noise-free barren plateaus,
which are linked to random parameter initialization. Our result is formulated
for a generic ansatz that includes as special cases the Quantum Alternating
Operator Ansatz and the Unitary Coupled Cluster Ansatz, among others. For the
former, our numerical heuristics demonstrate the NIBP phenomenon for a
realistic hardware noise model.
