Variational quantum algorithms rely on gradient based optimization to iteratively minimize a cost function evaluated by measuring output(s) of a quantum processor. A barren plateau is the phenomenon of exponentially vanishing gradients in sufficiently expressive parametrized quantum circuits. It has been established that the onset of a barren plateau regime depends on the cost function, although the particular behavior has been demonstrated only for certain classes of cost functions. Here we derive a lower bound on the variance of the gradient, which depends mainly on the width of the circuit causal cone of each term in the Pauli decomposition of the cost function. Our result further clarifies the conditions under which barren plateaus can occur.

变分量子算法依赖于基于梯度的优化来迭代地最小化通过测量量子处理器的输出来评估的成本函数。贫瘠的高原是在足够表达的参数化量子电路中梯度指数消失的现象。已经确定，贫瘠高原制度的开始取决于成本函数，尽管仅针对某些类别的成本函数证明了特定行为。这里我们推导出梯度方差的下界，主要取决于代价函数泡利分解中每一项的回路因果锥的宽度。我们的结果进一步阐明了可能出现贫瘠高原的条件。