The parameters of the quantum circuit in a variational quantum algorithm induce a landscape that contains the relevant information regarding its optimization hardness. In this work, we investigate such landscapes through the lens of information content, a measure of the variability between points in parameter space. Our major contribution connects the information content to the average norm of the gradient, for which we provide robust analytical bounds on its estimators. This result holds for any (classical or quantum) variational landscape. We validate the analytical understating by numerically studying the scaling of the gradient in an instance of the barren plateau problem. In such instance, we are able to estimate the scaling pre-factors in the gradient. Our work provides a way to analyze variational quantum algorithms in a data-driven fashion well-suited for near-term quantum computers.

变分量子算法中量子电路的参数会产生一个景观，其中包含有关其优化难度的相关信息。在这项工作中，我们通过信息内容的视角来研究此类景观，信息内容是参数空间中点之间变异性的度量。我们的主要贡献是将信息内容与梯度的平均范数联系起来，为此我们为其估计器提供了强大的分析界限。这个结果适用于任何（经典或量子）变分景观。我们通过数值研究贫瘠高原问题实例中梯度的缩放来验证分析的低估。在这种情况下，我们能够估计梯度中的缩放前置因子。我们的工作提供了一种以数据驱动的方式分析变分量子算法的方法，非常适合近期量子计算机。