Quantum Wavelet Transforms

The classical wavelet transform has been widely applied in the information processing field. It implies that quantum wavelet transform (QWT) may play an important role in quantum information processing. This chapter firstly describes the iteration equations of the general QWT using generalized tensor product. Then, Haar QWT (HQWT), Daubechies D4 QWT (DQWT), and their inverse transforms are proposed respectively. Meanwhile, the circuits of the two kinds of multi-level HQWT are designed. What's more, the multi-level DQWT based on the periodization extension is implemented. The complexity analysis shows that the proposed multi-level QWTs on 2n elements can be implemented by O(n3) basic operations. Simulation experiments demonstrate that the proposed QWTs are correct and effective.

Quantum Image Segmentation Algorithms

Quantum image segmentation has always been one of the difficult tasks in quantum image processing. This chapter introduce two quantum image segmentation algorithms. One is quantum edge detection algorithm; the other one is quantum image segmentation based on generalized Grover search algorithm.

Quantum Wavelet Packet Transforms

Quantum wavelet packet transform (QWPT) may play an important role in quantum information processing. In this chapter, the authors design quantum circuits of a generalized tensor product (GTP) and a perfect shuffle permutation (PSP). Next, they propose multi-level and multi-dimensional (1D, 2D and 3D) QWPTs, including Haar QWPT (HQWPT), D4 QWPT (DQWPT) based on the periodization extension and their inverse transforms for the first time, and prove the correctness based on the GTP and PSP. Furthermore, they analyze the quantum costs and the time complexities of the proposed QWPTs and obtain precise results. The time complexities of HQWPTs is at most six basic operations on 2n elements, which illustrates high efficiency of the proposed QWPTs.

Quantum Geometric Transformations

Geometric transformations are basic operations in image processing. This chapter describes geometric transformations of images and videos. These geometric transformations include two-point swapping, symmetric flip, local flip, orthogonal rotation, and translation.

The Storage and Retrieval Technologies of Quantum Images

Quantum image processing represents an emerging image processing technology by taking advantage of quantum computation. Quantum image processing faces the first question: How is an image stored in and retrieved from a quantum system? To solve the issue, the authors provide six quantum image representations, which can be divided into three categories. The first, second, and third categories store color information using amplitudes, phases, and basis states, respectively. Next, they design their circuits to implement the storage of quantum image. Then, retrieval methods are introduced. The storage and retrieval technologies of quantum image are the basis and premise condition to process quantum images.

Fundamentals of Quantum Computation

This chapter briefly describes the basic concepts and principles of quantum computing. Firstly, the concepts of qubit, quantum coherence, quantum decoherence, quantum entanglement, quantum density operators, linear operators, inner products, outer products, tensor products, Hermite operators, and unitary operators are described. Then, the four basic assumptions of quantum mechanics are introduced, focusing on the measurement assumptions of quantum mechanics. Finally, the definition of commonly used quantum logic gates is given including single qubit gates, double qubit gates, and multiple qubit gates. These contents provide the necessary theoretical basis for subsequent chapters.

Quantum Fourier Transforms

Quantum Fourier transform (QFT) plays a key role in many quantum algorithms, but the existing circuits of QFT are incomplete and lacking the proof of correctness. Furthermore, it is difficult to apply QFT to the concrete field of information processing. Thus, this chapter firstly investigates quantum vision representation (QVR) and develops a model of QVR (MQVR). Then, four complete circuits of QFT and inverse QFT (IQFT) are designed. Meanwhile, this chapter proves the correctness of the four complete circuits using formula derivation. Next, 2D QFT and 3D QFT based on QVR are proposed. Experimental results with simulation show the proposed QFTs are valid and useful in processing quantum images and videos. In conclusion, this chapter develops a complete framework of QFT based on QVR and provides a feasible scheme for QFT to be applied in quantum vision information processing.