Currently, the tensor completion problem has been paid high attention in the machine learning, especially in the field of computer vision and image processing. The low-rank tensor completion methods based on the tensor singular value decomposition and the tensor nuclear norm minimization has been proposed. However, they have limitations in computing speed, since they are SVD-based methods and need high computational cost for high dimensional tensor. In this paper, based on the tensor QR decomposition and the tensor nuclear norm, a fast low-rank tensor completion method is proposed. By reducing the dimensions of the tensor in the nuclear norm regularization term, the performance of the completion is substantially improved. Numerical experiments for color images, MRI and videos demonstrate that the effectiveness of the proposed method.