Inspired by complexity and diversity of biological neurons, our group proposed quadratic neurons by replacing the inner product in current artificial neurons with a quadratic operation on input data, thereby enhancing the capability of an individual neuron. Along this direction, we are motivated to evaluate the power of quadratic neurons in popular network architectures, simulating human-like learning in the form of quadratic-neuron-based deep learning. Our prior theoretical studies have shown important merits of quadratic neurons and networks in representation, efficiency, and interpretability. In this paper, we use quadratic neurons to construct an encoder-decoder structure, referred as the quadratic autoencoder, and apply it to low-dose CT denoising. The experimental results on the Mayo low-dose CT dataset demonstrate the utility of quadratic autoencoder in terms of image denoising and model efficiency. To our best knowledge, this is the first time that the deep learning approach is implemented with a new type of neurons and demonstrates a significant potential in the medical imaging field.