Noise-Induced Barren Plateaus in Variational Quantum Algorithms

Variational Quantum Algorithms (VQAs) may be a path to quantum advantage on Noisy Intermediate-Scale Quantum (NISQ) computers. A natural question is whether the noise on NISQ devices places any fundamental limitations on the performance of VQAs. In this work, 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 layers $L$. This implies exponential decay in the number of qubits $n$ when $L$ scales as $\operatorname{poly}(n)$, for sufficiently large coefficients in the polynomial. 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 an abstract ansatz that includes as special cases the Quantum Alternating Operator Ansatz (QAOA) and the Unitary Coupled Cluster Ansatz, among others. In the case of the QAOA, we implement numerical heuristics that confirm the NIBP phenomenon for a realistic hardware noise model.