The optimization of quantum circuits can be hampered by a decay of average gradient amplitudes with increasing system size. When the decay is exponential, this is called the barren plateau problem. Considering explicit circuit parametrizations (in terms of rotation angles), it has been shown in Arrasmith et al (2022 Quantum Sci. Technol. 7 045015) that barren plateaus are equivalent to an exponential decay of the variance of cost-function differences. We show that the issue is particularly simple in the (parametrization-free) Riemannian formulation of such optimization problems and obtain a tighter bound for the cost-function variance. An elementary derivation shows that the single-gate variance of the cost function is strictly equal to half the variance of the Riemannian single-gate gradient, where we sample variable gates according to the uniform Haar measure. The total variances of the cost function and its gradient are then both bounded from above by the sum of single-gate variances and, conversely, bound single-gate variances from above. So, decays of gradients and cost-function variations go hand in hand, and barren plateau problems cannot be resolved by avoiding gradient-based in favor of gradient-free optimization methods.

中文翻译：

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量子电路的成本集中和梯度消失的等价：黎曼公式的基本证明


随着系统尺寸的增加，平均梯度幅度的衰减可能会阻碍量子电路的优化。当衰减呈指数级时，这称为贫瘠高原问题。考虑到显式电路参数化（就旋转角度而言），Arrasmith 等人 (2022 Quantum Sci. Technol.7 045015) 表明，贫瘠平台相当于成本函数差异方差的指数衰减。我们表明，在此类优化问题的（无参数化）黎曼公式中，该问题特别简单，并且获得了成本函数方差的更严格界限。初等推导表明，成本函数的单门方差严格等于黎曼单门梯度方差的一半，其中我们根据均匀哈尔测度对变量门进行采样。然后，成本函数及其梯度的总方差都从上方以单门方差之和为界，反之亦然，从上方以单门方差为界。因此，梯度衰减和成本函数变化是相辅相成的，并且不能通过避免基于梯度而支持无梯度优化方法来解决贫瘠平台问题。