Fig. 3: QAOA heuristics in the presence of realistic hardware noise: increasing number of rounds for fixed problem size.

a The approximation ratio averaged over 100 random graphs of 5 nodes is plotted versus number of rounds p. The black, green, and red curves respectively correspond to noise-free training, noisy training with noise-free final cost evaluation, and noisy training with noisy final cost evaluation. The performance of noise-free training increases with p, similar to the results in Ref. ¹⁵. The green curve shows that the training process itself is hindered by noise, with the performance decreasing steadily with p for p > 4. The dotted blue lines correspond to known lower and upper bounds on classical performance in polynomial time: respectively the performance guarantee of the Goemans-Williamson algorithm⁷⁷ and the boundary of known NP-hardness^78,79. b The deviation of the cost from \({{{{{{{\rm{Tr}}}}}}}}[{H}_{P}]/{2}^{n}\) (averaged over graphs and parameter values) is plotted versus p. As p increases, this deviation decays approximately exponentially with p (linear on the log scale). c The absolute value of the largest partial derivative, averaged over graphs and parameter values, is plotted versus p. The partial derivatives decay approximately exponentially with p, showing evidence of Noise-Induced Barren Plateaus (NIBPs).