| Quantum random walks (QWs), as the quantum version of classical random walks, since was introduced in1993, have been widespread concerned. Recently, QWs have attracted great attention from mathematicians, computer scientists, physicists, and engineers. Quantum walks have been exploited as an useful tool for quantum algorithms in quantum computing. A list of quantum algorithms based on QWs have already been proposed. QWs can be used to perform an oracle search on a database of N items with O((?)N) was proved, and in some problems, the hitting time is exponential faster than its corresponding classical random walk, QWs can also be used for universal computation.In order to use the quantum walk to its fullest potential, we need to have a deeper understanding of the basic nature, including standard deviation, entropy, hitting time, mixing time and average position etc., we need more precise results about them and QWs in more situations need to be discussed.In this dissertation, we mainly concentrated some basic properties and appli-cations of QWs. It contains,1. We give the definition of quantum random walk, and the basic difference between classical random walk and quantum random walk. Two methods for computing one-dimensional QWs:Fourier transform method and per-mutations method were introduced. Using3parameters unitary operation to control the evolution of QWs, some algorithms, the solution of universal quantum computing and the evolution of multi-particles in QWs were also introduced,2. We investigated discrete-time quantum walks with an arbitary unitary coin. Here we discover that the average position (x)=max((x))sin(α+γ), while the initial state is1/(?)2(|OL)+i|OR)). We prove the result and get some symmetry properties of quantum walks with a U(2) coin with|OL) and|OR) as the initial state.3. We investigated the discrete-time quantum random walks on a line in peri-odic potential. The probability distribution with periodic potential is more complex compared to the normal quantum walks, and the standard devia-tion a has interesting behaviors for different period q and parameter θ. We studied the behavior of standard deviation with variation in walk steps, pe-riod, and θ. The standard deviation increases approximately linear with θ and decreases with1/q for θ∈(0, π/4), and increases approximately linear with1/q for θ∈[π/4, π/2). For θ∈(π/4, π/2), it means the transmission is larger than the reflection, and become larger, with sensibility, the diffussion will be accelerated, and when θ∈(π/2,3π/4), the transmission becomes smaller and reflection becomes larger, with sensibility, the diffusion will be decelerated, but when θ∈(π/4,3π/4) and q=2, the standard deviations are nearly the same, and when and q>2, the standard deviation will decrease.4. We construct a Parrondo’s game using discrete time quantum walks. Two lossing games are represented by two different coin operators. By mixing the two coin operators UA(αA,βA,γA) and UB(αB,βB.γB), we may win the game. Here we mix the two games correlated to the positions instead of time. With a number of selections of the parameters, we can win the game with sequences ABB, ABBB, etc.. If we set βA=45°,γA=0, αB=0,βB=88°, we find the game1with Uas=Us(-51°,45°,0), UBS=Us(0,88°,-16°) will win and get the most profit. If we set αA=0,βA=45°, αB=0,βB=88°and the game2with UAS=Us(0,45°,-51°), UBS=Us(0,88°,-67°), will win most. And game1is equivalent to the game2with the changes of sequences and steps. But at a large enough steps, the game will loss at last. |
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