In variational quantum algorithms the parameters of a parameterized quantum circuit are optimized in order to minimize a cost function that encodes the solution of the problem. The barren plateau phenomenon manifests as an exponentially vanishing dependence of the cost function with respect to the variational parameters, and thus hampers the optimization process. We discuss how, and in which sense, the phenomenon of noise-induced barren plateaus emerges in parameterized quantum circuits with a layered noise model. Previous results have shown the existence of noise-induced barren plateaus in the presence of local Pauli noise (Wang et al 2021 Nat. Commun. 12 6961). We extend these results analytically to arbitrary completely-positive trace preserving maps in two cases: (1) when a parameter-shift rule holds, (2) when the parameterized quantum circuit at each layer forms a unitary 2-design. The second example shows how highly expressive unitaries give rise not only to standard barren plateaus (McClean et al 2018 Nat. Commun. 9 4812), but also to noise-induced ones. In the second part of the paper, we study numerically the emergence of noise-induced barren plateaus in QAOA circuits focusing on the case of MaxCut problems on d-regular graphs and amplitude damping noise.

中文翻译：

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任意分层噪声模型中噪声引起的贫瘠高原的出现


在变分量子算法中，参数化量子电路的参数被优化，以便最小化编码问题解决方案的成本函数。贫瘠高原现象表现为成本函数对变分参数的指数消失依赖性，从而阻碍了优化过程。我们讨论了噪声引起的贫瘠平台现象如何以及在何种意义上出现在具有分层噪声模型的参数化量子电路中。先前的结果表明，在存在局部泡利噪声的情况下，存在噪声引起的贫瘠高原（Wang等人2021年纳特。交流。 12 6961）。我们将这些结果分析性地扩展到两种情况下的任意完全正迹保留映射：（1）当参数移位规则成立时，（2）当每层的参数化量子电路形成单一2-设计时。第二个例子展示了高度表现力的单一性如何不仅产生标准的贫瘠高原（麦克林等人2018年纳特。交流。 9 4812），也适用于噪声引起的。在本文的第二部分中，我们对 QAOA 电路中噪声引起的贫瘠平台的出现进行了数值研究，重点关注 MaxCut 问题的情况d - 常规图表和振幅阻尼噪声。