Barren Plateau Issues for Variational Quantum-Classical Algorithms

Variational quantum-classical algorithms (VQCAs) optimize the parameters of a quantum neural network, V, to minimize a cost function, C. Many researchers believe that VQCAs will enable the first practical applications of noisy quantum computers. However, VQCAs are heuristic methods with unproven scaling. Here, we rigorously prove two results related to the trainability of VQCAs. Our first result states that choosing C to be a global function of V leads to an exponentially vanishing gradient (i.e., a barren plateau) even when V is shallow. This implies that many VQCAs proposed in the literature must revise their proposed cost functions. Our second results states that, under the same conditions, choosing C to be a local function of V leads to a non-vanishing gradient, i.e., with the gradient vanishing no worse than polynomially. This suggests that VQCAs have the potential to be trainable, if one chooses an appropriate cost function. We support these analytical results with numerics for large problem sizes.