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.

中文翻译：

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贫瘠高原对无梯度优化的影响

贫瘠的高原景观对应于量子比特数呈指数级消失的梯度。这种景观已经在变分量子算法和具有深度电路或全局成本函数的量子神经网络中得到了证明。出于显而易见的原因，预计基于梯度的优化器将受到贫瘠高原的显着影响。然而，无梯度优化器是否受到影响是一个有争议的话题，一些人认为无梯度方法不受贫瘠高原的影响。在这里，我们表明，事实上，无梯度优化器并不能解决贫瘠的高原问题。我们的主要结果证明，作为无梯度优化决策基础的成本函数差异在贫瘠的平台上被指数抑制。因此，没有指数精度，无梯度优化器不会在优化中取得进展。我们通过在一个贫瘠的平台上用几个无梯度优化器（Nelder-Mead、Powell 和 COBYLA 算法）进行训练，在数值上证实了这一点，并表明优化中所需的镜头数量随着量子比特的数量呈指数增长。