Quantum neural networks (QNNs) offer a powerful paradigm for programming
near-term quantum computers and have the potential to speedup applications
ranging from data science to chemistry to materials science. However, a
possible obstacle to realizing that speedup 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
quantum neural networks (outside of the context of BPs).

本研究探讨了量子神经网络中的 Barren Plateau 现象，研究发现高阶导数信息无法解决 Barren Plateau 的指数级灭减问题，因此超越（一阶）梯度下降的优化策略仍会受到 Barren Plateau 的影响。同时，我们证明了新颖的通用公式，可用于在量子硬件上分析求解任何高阶偏导数，这些公式可能有独立的兴趣和用途，可用于量子神经网络的训练（除了 Barren Plateau 的情境）。