Recently with the rapid development of technology, there are a lot of applications require to achieve low-cost learning in order to accomplish inexpensive computation. However the known computational power of classical artificial neural networks (CANN), they are not capable to provide low-cost learning due to many reasons such as linearity, complexity of architecture, etc. In contrast, quantum neural networks (QNN) may be representing a good computational alternate to CANN, based on the computational power of quantum bit (qubit) over the classical bit. In this paper, a new algorithm of quantum perceptron neural network based only on one neuron is introduced to overcome some limitations of the classical perceptron neural networks. The proposed algorithm is capable to construct its own set of activation operators that enough to accomplish the learning process in a limited number of iterations and, consequently, reduces the cost of computation. For evaluation purpose, we utilize the proposed algorithm to solve five problems using real and artificial data. It is shown throughout the paper that promising results are provided and compared favorably with other reported algorithms