In variational quantum algorithms, it is important to balance conflicting requirements of expressibility and trainability of a parameterized quantum circuit (PQC). However, appropriate PQC designs are not necessarily trivial. Here, we propose an algorithm for optimizing the PQC structure, where single-qubit gates are sequentially replaced by the optimal ones via diagonalization of a matrix whose elements are evaluated on slightly modified circuits. This replacement leads to a better approximation of target states with limited circuit depth. Furthermore, we clarify the existence of a barren plateau in the sequential optimization in terms of the spectrum concentration of the matrix, which defines the cost landscape with respect to changes in the target gate. Then, we rigorously show the concentration is no faster than polynomials in the number of qubits when an n-qubit PQC depth is O(log⁡n) using local observables. Finally, numerical experiments are provided to show the convergence of our method which is faster than classical optimizers on both simulators and a real device. Our results provide evidences for sequential optimizers as better alternatives to optimize PQCs on near-term quantum devices.

中文翻译：

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参数化量子电路中单量子位门的顺序最优选择


在变分量子算法中，平衡参数化量子电路（PQC）的可表达性和可训练性的相互冲突的要求非常重要。然而，适当的 PQC 设计并不一定是微不足道的。在这里，我们提出了一种优化 PQC 结构的算法，其中通过对矩阵进行对角化，将单量子位门顺序替换为最佳门，该矩阵的元素在稍微修改的电路上进行评估。这种替换可以在有限的电路深度下更好地逼近目标状态。此外，我们根据矩阵的频谱集中阐明了顺序优化中存在的贫瘠平台，它定义了相对于目标门变化的成本格局。然后，我们使用局部可观测量严格证明，当 n 量子位 PQC 深度为 O(log⁡n) 时，集中度并不比量子位数量的多项式快。最后，提供了数值实验来证明我们的方法的收敛性，该方法在模拟器和真实设备上都比经典优化器更快。我们的结果为顺序优化器作为优化近期量子设备上的 PQC 的更好替代方案提供了证据。