This paper provides a new adaptive algorithm for single-step prediction by modeling the potential field of a one di-mension Schr¨odinger wave equation using neural network. This new architecture is referred to as the recurrent quantum neural network (RQNN). The RQNN can filter the signal embedded with non-stationary noise without any priori knowledge of the shape of the signal and statistical properties of the noise. We compared the simulation results of the RQNN with a classical adaptive stochastic filter-RLS. It is shown that the RQNN is much more efficient in denoising signals embedded with Gaus-sian stationary, non-Gaussian stationary and Gaussian non-stationary noise such as DC, sinusoid, staircase and speech sig-nals. The RQNN can enhance the signal to noise rate (SNR) by 20 dB, which is more than 10 dB given by the traditional technology when it denoising sinusoid signal.