Application Of Quantum Walks In Analyzing Physical System's Topological Signatures And Relativistic Effects

Random walk is a powerful tool for designing efficient implementation of algorithms such as crypotographyic schemes.Quantum walks are the quantum correspondence of classical random walks. Because of the special properties of quantum mechanics including Heisenberg uncertainty principle, no-cloning theorem for an unknown state and the reliable indistinguishability attribute of non-orthogonal quantum states, compared with classical random walks there are some unique properties appeared in the evolutions of quantum walks. As a foundation of designing new quantum algorithms, quantum walks have a huge potiential in the quantum private communication and quantum information processing. Quantum walks are also powerful platforms for simulating and analyzing physical systems. This thesis considers the applications of quantum walks in analyzing physical systems' topological signatures and relativistic effects and their implementation schemes.The application of quantum walks in analyzing physical systems'topological signatures: Since the discovery of quantum Hall effect,topological properties of quantum physics system has aroused widespread concern on the theoretical and experimental. It was discovered that the quantum integers, fractions Hall effect and topological properties of topological insulators, topological computations based on the topological properties of physical systems.But in the experiment, the measurement of the physical system's topology order is very difficult, often requiring powerful magnetic field and low temperature conditions. In this work, we demonstrate that continuous-time quantum walks can be used to define topological phase order and topological transitions in spin systems. For this purpose, we consider a two-dimensional spin system with arbitrary spin-orbit coupling Hamiltonian H. In two-dimensional continuous-time quantum walks, the particle's evolution is governed by the Hamiltonians having non-trival topological signatures. This dynamical evolution is related to the topological order of the Hamiltonian. Governed with Hamiltonians having different Chern numbers, the spreading speed and the evolution pattern of the pariticle will be different and they will change at exactly the phase transition point of the Hamiltonians. Specifically, the first order time derivative of the standard deviation of the walker's probability distribution in position space, where the quantum walk evolution is governed by the Hamiltonian, changes at the point of a topological phase transition, this physical variant in quantum walks can be used to detect the topological order of physical systems.The relativistic effects in quantum walks: In 2006, Strauch analyzed the relation between the one-dimensional Dirac equation and quantum walk for the first time, and noted that the diffusion of the localized initial state has the similar time evolution with a relativistic particle density wave packet, i.e. bimodal probability distribution. After analysis we found that the two peaks of the wave packet can not be characterized as a relativistic effect in discrete-time quantum walk. In discrete-time quantum walks, when the initial sate is delocalized state(the probability distribution is Gaussian distribution) and the parameter in the coin operator is small values, the effective Hamiltonians of this evolution have the formula of Dirac Hamiltonians and their eigenvalues have the linear dispersion relation. Under this situation, the particle will evolve into aSchoridinger cat states. When the paramenter in coin operator is large, with delocalized states as initial sates, we find that there will also be Schoridinger cat states appearing. Under this situation,the effective Hamiltonians are not Dirac Hamiltonians which include the quadratic terms of momentum operator. But the eigenvalues still have the linear dispersion relations and the corresponding eigensstates weakly dependent on momentum. Therefore, we conclude that the linear dispersion cannot be a good critia to classify the relativistic physical systems in quantum walks.Universal quantum walks optical implementations: we propose an optical setup for generalized quantum walks-electric quantum walks and split-step quantum walks. Our setup works in spin space and orbital angular momentum space of light which makes our scheme scalable and efficient. We give the implementation scheme to prepare, store cat states and adjust its size, which pays an impact for quantum algorithms,computation and communication based on cat states. Our implementation scheme of split-step quantum walks can also be used for demonstrating topological edge states.