Quantum Discrete Cosine Transform for Image Compression
  Discrete Cosine Transform (DCT) is very important in image compression. Classical 1-D DCT and 2-D DCT has time complexity O(NlogN) and O(N&sup2;logN) respectively. This paper presents a quantum DCT iteration, and constructs a quantum 1-D and 2-D DCT algorithm for image compression by using the iteration. The presented 1-D and 2-D DCT has time complexity O(sqrt(N)) and O(N) respectively. In addition, the method presented in this paper generalizes the famous Grover's algorithm to solve complex unstructured search problem. 
Chao Yang Pang, Zheng Wei Zhou, Guang Can Guo