Study On The Dynamic Behavior Of Wave Packet Diffusion In Non - Uniform Quantum Random Walk

Recently, with the demand for high speed, secrecy, the large capacity of communi-cation and computation of our society, theory and experimental researches in quantum information and quantum computation has achieved in-depth development. Quantum algorithm shows that quantum computer prior to the classic computer computing funda-mentally. Designing a good and suitable quantum algorithm for quantum computing has become an important topic in quantum computing theory, quantum random walk has drawn extensive attention as the counterpart of classical random walk and a natural ex-tension of quantum systems. This is because classical random walk has wide applications in traditional algorithms. Current studies have shown that the quantum random walk based algorithms can effectively speed up the data processing and calculation process. So the research of quantum random walk has important significance for designing new quantum algorithm. In addition, as a kind of relatively simple physical model, quantum random walk has very rich connotation. It has profound significance for the control of atoms in the physical system, microscopic system such as photons movement by studying the dynamics of quantum random walk.The quantum random walk can be seen as a particle that has two freedoms(the position space freedom and the spin space freedom). The walking process can be seen as the evolution of the wave function of the particle. The evolution operator consists of two parts:the C matrix that acts on and rotates the spin state and the S matrix that determines the walking direction of the particle according to its spin state. The C matrix is determined by a parameter θ. When θ do not change with time or space, we call that homogeneous quantum random walk. When θ changes with time or space, we call it inhomogeneous quantum random walk. We numerically study the dynamical behavior of one-dimensional time aperiodic and space aperiodic quantum random walk. We discuss the spread behavior of the wave packet that when the parameter6takes two different values α.β and they change aperiodically with time or space.Firstly, we discuss the wave function evolutes with generalized Fibonacci sequence in time. It is found that the initial state impacts greatly on the position probability diffusion of the particle and the spreading behavior of the wave packet is sub-ballistic, that is the standard deviation is σ(t)～tγ,γ€(0,1), for both classes of generalized Fibonacci sequences. The exponent γ can be changed with the changing of the parameters α,β. For the first class of generalized Fibonacci sequence, the exponent γ is ranged in0.5and1. While for the second class of generalized Fibonacci sequence, the exponent γ ranged in0.9and1and it is larger than that of the first class of generalized Fibonacci sequence. From the position probability distribution and the standard deviation of the two classes of generalized Fibonacci sequence, we can see that for the first class of generalized Fibonacci sequence, the spreading behavior of the wave packet is closer to the classical random walk(γ=0.5), while for the second class of generalized Fibonacci sequence, it is closer to the homogeneous quantum random walk(γ=1).Secondly, we study the wave function evolutes with generalized Fibonacci sequence in space. It is found that the initial state impacts differently on the position probability diffusion in the two classes of generalized Fibonacci sequences. For the first class of generalized Fibonacci sequence, it impacts little. While for the second class of generalized Fibonacci sequence, it impacts a lot. The wave packet is sub-ballistic spreading for both of the two classes of generalized Fibonacci sequences. The exponent γ can be changed with the changing of the parameters a,/3. For the first class of generalized Fibonacci sequence, the exponent γ is ranged in0and1. While for the second class of generalized Fibonacci sequence, most values of the exponent γ can be ranged in0.8and1and it is larger than that of the first class of generalized Fibonacci sequence. Like the time aperiodic quantum random walk, for the first class of generalized Fibonacci sequence, the spreading behavior of the wave packet is closer to the classical random walk, while for the second class of generalized Fibonacci sequence, it is closer to the homogeneous quantum random walk. In contrast, the wave packet is localized for the random sequences. The standard deviation tends to a constant with the increase of time and the width of the wave packet will not increase over the time.