Signal and image compression using quantum discrete cosine transform

The discrete cosine transform (DCT) is widely used in image and video compression standard formats. This is due to its ability to represent signals and images using a limited number of significant coefficients without noticeable loss of visual clarity. The classical one-dimensional discrete cosine transform (1D - DCT) and two-dimensional discrete cosine transform (2D - DCT) have computational complexities of O(Nlog2N) and O(N2log2N), respectively. Thus, as the images grow in size, the runtime of the DCT highly increases which could limit its usability in real-time applications. This paper presents a quantum DCT algorithm (QDCT) that is more efficient than its classical counterpart in terms of complexity. Furthermore, the proposed QDCT is used to develop and realize a quantum image compression technique. The developed compression technique performs a search to determine the most significant computed DCT coefficients and is derived from Grover's algorithm. It provides a generalization to the original search algorithm by utilizing two oracle operators to solve the complex unstructured search problem rather than a single one. Thus, the proposed QDCT algorithm can simultaneously calculate the DCT coefficients and identify the significant DCT coefficients through applying two oracles. The comparison of the introduced QDCT with Grover's algorithm also indicates that the QDCT algorithm is more efficient. This can be attributed to performing a rotation on the subspace rather than on the global space in Grover's algorithm. In addition, the presented quantum 1D - and 2D - DCT have reduced complexities compared to the classical algorithms which are O(N) and O(N), respectively. Therefore, the presented QDCT and compression algorithm can be applied efficiently to accomplish various transform-based quantum signal and image processing tasks.