The advent of deep-learning technology promises major leaps forward in addressing the ever-enduring problems of wireless resource control and optimization, and improving key network performances, such as energy efficiency, spectral efficiency, transmission latency, etc. Therefore, a common understanding for quantum deep-learning algorithms is that they exploit advantages of quantum hardware, enabling massive optimization speed ups, which cannot be achieved by using classical computer hardware. In this respect, this paper investigates the possibility of resolving the energy efficiency problem in wireless communications by developing a quantum neural network (QNN) algorithm of deep-learning that can be tested on a classical computer setting by using any popular numerical simulation tool, such as Python. The computed results show that our QNN algorithm can be indeed trainable and that it can lead to solution convergence during the training phase. We also show that the proposed QNN algorithm exhibits slightly faster convergence speed than its classical ANN counterpart, which was considered in our previous work. Finally, we conclude that our solution can accurately resolve the energy efficiency problem and that it can be extended to optimize other communications problems, such as the global optimal power control problem, with promising trainability and generalization ability.