  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 {CNOT, Rz},
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.
