The Algorithms And Simulation Of Multi-qubit Quantum Wavelet Transforms

Quantum computing is a recently developed subject. It can deal with information technology using quantum mechanics. With the growth of the theory and the technology, more experts and scholars join in the research of the field; at the same time, quantum computing grows rapidly and promotes the development and the progress of the computer science. In the not-far future, our society will step into quantum information times. For the recent 20 years, quantum computing theory has shown that it is much superior to classic computing in many ways. Especially, the superiority is more evident in some simulation of quantum system and large number factoring problems etc. Meantime, many quantum algorithms are presented. At the other hand, with the development of the theory of wavelet research and extensive application of wavelet analysis in the field of signal analysis and image processing, wavelet analysis has been thought more and more important in the field of quantum computing. In this paper, we introduce the algorithms of quantum Haar wavelet and quantum DaubechiesD⁴wavelet transform, which are based on the quantum Fourier transform. We present the design of 3-qubit pulse sequences of quantum wavelet transform employing the NMR physical technology with the apprehending and applying the multiple-quantum operator algebra spaces and the NMR technology. Furthermore, we verify whether our design is true, reasonable and feasible using QCE. In the total course of designing NMR pulse sequences, the key to problems lies in how to translate unitary transformation corresponding to quantum wavelet transform algorithms into a series of 1-qubit and 2-qubit quantum gates products, in particular, how to exactly decompose the inter-nuclear interaction as a sequence of a limited number of one- and two-qubit gates.Chapter 1: in the beginning of the paper, we introduce the history of quantum computing and its status today, and we describe the method, content and meaning of our research.Chapter 2: we introduce quantum computing. From several facets, such as qubits, quantum logic gates and parallel computing etc, we present quantum computing.Chapter 3: we discuss the quantum wavelet transform algorithms, and we present their 3-qubit quantum logical circuit and the complexity.Chapter 4: we introduce the multiple-quantum operator algebra spaces. We describe the decomposition of unitary transformation and analysis the feature of the elementary quantum circuits.Chapter 5: we introduce the NMR technology in detail, and bring forward the system and basic theory methods of quantum computing.Chapter 6: we present the NMR pulse sequence design of 1-qubit quantum logical gates and 2-qubit CNOT gate. Then we provide the NMR pulse sequence design of 3-qubit quantum Haar wavelet and quantum DaubechiesD⁴ wavelet transform algorithms. At last we simulate them with QCE.Chapter 7: summarize and forecast the quantum computing.