论文标题
成本功能依赖于贫瘠的贫瘠高原，浅参数化量子电路
Cost Function Dependent Barren Plateaus in Shallow Parametrized Quantum Circuits
论文作者
论文摘要
变性量子算法（VQAS）优化参数化量子电路的参数$ \vecθ$ $ v（\vecθ）$，以最大程度地降低成本函数$ c $。尽管VQA可以实现嘈杂的量子计算机的实际应用，但它们仍然是未经证实的缩放的启发式方法。在这里，我们严格证明了两个结果，假设$ v（\vecθ）$是由形成本地2设计的块组成的交替分层ansatz。我们的第一个结果指出，即使$ v（\vecθ）$浅，定义$ c $的全球可观察结果都会导致梯度消失（即贫瘠的高原）。因此，文献中的几个VQA必须修改其拟议费用。另一方面，我们的第二个结果指出，定义$ c $使用本地可观察物的定义会导致最坏的是多项式消失的梯度，只要$ v（\vecθ）$的深度为$ \ nathcal {o}（\ log log n）$。我们的结果建立了区域和训练性之间的联系。我们通过量子自动编码器实现的大规模模拟（最多100 QUAT）来说明这些想法。
Variational quantum algorithms (VQAs) optimize the parameters $\vecθ$ of a parametrized quantum circuit $V(\vecθ)$ to minimize a cost function $C$. While VQAs may enable practical applications of noisy quantum computers, they are nevertheless heuristic methods with unproven scaling. Here, we rigorously prove two results, assuming $V(\vecθ)$ is an alternating layered ansatz composed of blocks forming local 2-designs. Our first result states that defining $C$ in terms of global observables leads to exponentially vanishing gradients (i.e., barren plateaus) even when $V(\vecθ)$ is shallow. Hence, several VQAs in the literature must revise their proposed costs. On the other hand, our second result states that defining $C$ with local observables leads to at worst a polynomially vanishing gradient, so long as the depth of $V(\vecθ)$ is $\mathcal{O}(\log n)$. Our results establish a connection between locality and trainability. We illustrate these ideas with large-scale simulations, up to 100 qubits, of a quantum autoencoder implementation.
