Quantum neural networks (QNNs) offer a powerful paradigm for programming near-term quantum computers and have the potential to speed up applications ranging from data science to chemistry to materials science. However, a possible obstacle to realizing that speed-up is the barren plateau (BP) phenomenon, whereby the gradient vanishes exponentially in the system size n for certain QNN architectures. The question of whether high-order derivative information such as the Hessian could help escape a BP was recently posed in the literature. Here we show that the elements of the Hessian are exponentially suppressed in a BP, so estimating the Hessian in this situation would require a precision that scales exponentially with n. Hence, Hessian-based approaches do not circumvent the exponential scaling associated with BPs. We also show the exponential suppression of higher order derivatives. Hence, BPs will impact optimization strategies that go beyond (first-order) gradient descent. In deriving our results, we prove novel, general formulas that can be used to analytically evaluate any high-order partial derivative on quantum hardware. These formulas will likely have independent interest and use for training QNNs (outside of the context of BPs).

量子神经网络 (QNN) 为近期量子计算机编程提供了强大的范例，并有可能加速从数据科学到化学再到材料科学的应用。然而，实现加速的一个可能障碍是贫瘠高原 (BP) 现象，即对于某些 QNN 架构，梯度在系统大小n 中呈指数消失。最近在文献中提出了诸如 Hessian 之类的高阶导数信息是否有助于逃避 BP 的问题。在这里，我们展示了 Hessian 的元素在 BP 中被指数抑制，因此在这种情况下估计 Hessian 需要一个与n 成指数缩放的精度. 因此，基于 Hessian 的方法并没有规避与 BP 相关的指数缩放。我们还展示了高阶导数的指数抑制。因此，BP 将影响超越（一阶）梯度下降的优化策略。在推导我们的结果时，我们证明了新颖的通用公式，可用于分析评估量子硬件上的任何高阶偏导数。这些公式可能对训练 QNN 具有独立的兴趣和用途（在 BP 的上下文之外）。