Performances and limitations of variational quantum algorithms under realistic noise models

Variational quantum algorithms (VQAs) are promising candidates to perform useful tasks with the currently available noisy quantum computers. However, these algorithms may suffer from the problem of barren plateaus that hinder their trainability. Recently, it was also shown that general Pauli noise can worsen the barren plateau problem [1].

We consider the performances of VQAs under more realistic noise models, like dephasing and/or amplitude-damping noise, focusing on the Quantum Approximate Optimization Algorithm (QAOA) for MaxCut problems. We show how sampling over random circuit parameters, and in the limit of weak noise, the output state of the quantum circuit approaches with high probability the completely mixed state exponentially in the circuit depth. This, in turn, implies exponential decay also of the gradient of the cost function. We show how this exponential decay can be explained as a unitary 2-design property of the QAOA circuit, which effectively turns the noise into a global depolarizing one. While it is known that some QAOA circuits give rise to universal unitaries [2], we show numerically that for our examples the noise is well described by the global depolarizing channel, even for single parameter sets.