Optimized Synthesis of Circuits for Diagonal Unitary Matrices with Reflection Symmetry

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.