Barren plateau landscapes correspond to gradients that vanish exponentially
in the number of qubits. Such landscapes have been demonstrated for variational
quantum algorithms and quantum neural networks with either deep circuits or
global cost functions. For obvious reasons, it is expected that gradient-based
optimizers will be significantly affected by barren plateaus. However, whether
or not gradient-free optimizers are impacted is a topic of debate, with some
arguing that gradient-free approaches are unaffected by barren plateaus. Here
we show that, indeed, gradient-free optimizers do not solve the barren plateau
problem. Our main result proves that cost function differences, which are the
basis for making decisions in a gradient-free optimization, are exponentially
suppressed in a barren plateau. Hence, without exponential precision,
gradient-free optimizers will not make progress in the optimization. We
numerically confirm this by training in a barren plateau with several
gradient-free optimizers (Nelder-Mead, Powell, and COBYLA algorithms), and show
that the numbers of shots required in the optimization grows exponentially with
the number of qubits.

该文研究了关于到底梯度自由优化器对于荒芜高原问题是否受影响存在争议，通过数值实验和理论分析，发现在荒芜高原情况下，梯度自由优化器无法解决问题，因为成本函数差异在荒芜高原被指数抑制，使得梯度自由优化器无法在优化过程中取得进展。