During the noisy intermediate-scale quantum (NISQ) era, it is important to optimize the quantum circuits in circuit depth and gate count, especially entanglement gates, including the CNOT gate. Among all the unitary operators, diagonal unitary matrices form a special class that plays a crucial role in many quantum algorithms/subroutines. Based on a natural gate set \(\{ \text{CNOT},R_{z}\} \), quantum circuits for general diagonal unitary matrices were discussed in several previous works, and an optimal synthesis algorithm was proposed in terms of circuit depth. In this paper, we are interested in the implementation of diagonal unitary matrices with reflection symmetry, which has promising applications, including the realization of real-time evolution for first quantized Hamiltonians by quantum circuits. Owing to such a symmetric property, we show that the quantum circuit in the existing work can be further simplified and propose a constructive algorithm that optimizes the entanglement gate count. Compared to the previous synthesis methods for general diagonal unitary matrices, the quantum circuit by our proposed algorithm achieves nearly half the reduction in both the gate count and circuit depth.

中文翻译：

---

反射对称对角酉矩阵电路的优化综合

在噪声中尺度量子（NISQ）时代，优化量子电路的电路深度和门数非常重要，特别是纠缠门，包括CNOT门。在所有酉算子中，对角酉矩阵构成了一个特殊的类别，在许多量子算法/子程序中发挥着至关重要的作用。基于自然门集\(\{ \text{CNOT},R_{z}\} \)，在之前的几篇工作中讨论了一般对角酉矩阵的量子电路，并在电路深度方面提出了一种最优合成算法。在本文中，我们感兴趣的是具有反射对称性的对角酉矩阵的实现，它具有广阔的应用前景，包括通过量子电路实现第一量化哈密顿量的实时演化。由于这种对称性，我们表明现有工作中的量子电路可以进一步简化，并提出了一种优化纠缠门数量的建设性算法。与之前的一般对角酉矩阵的合成方法相比，我们提出的算法的量子电路在门数和电路深度上都实现了近一半的减少。