Utilizing Quantum States:the Study Of Quantum Random Walk On Graphs

As is known,quantum computation may help human beings to overcome the obstacle faced by classical computational technologies.In order to study quantum computation,people develop many powerful tools,one of which is quantum walk.The quantum walk is proposed as the counterpart of classical random walks.In this thesis,we study the.quantum random walks on typical graphs and the quantum gates based on quantum walk systems as well as their decoherence.In the first chapter,we introduce the background of quantum walk and several mathematical models of quantum walk systems.We also review some quantum-walk based quantum algorithms and some recent studies on the universality of quantum walks.Additionally,we introduce some experiment for quantum-walk systems.In the second chapter,we study the quantum walk on the triple type graph with the help of Green function method.We find the parity of the side chain's length can directly influence the propagation feature of the particle on main chain.We confirm these analytical results by numerical simulation.In order to understand these phenomenon,we give a physical explanation.Besides,we propose a experimental scheme of such a quantum-walk system.In the third chapter,we study the quantum walk on a type of graphes which has an Abelian symmetry.We show these quantum-walk systems can construct several effective quantum-gate systems,such as the X-gate of a single qubit and the control gate of two qubits.We also propose two experimental schemes of such quantum-walk systems.In the forth chapter,we study the decoherence of the above mentioned effec-tive quantum-gate systems under noise environment.After considering adiabatic approximation,we find the decoherence behaviour is closely related with the corre-lation function of the noises.We consider some typical spectrums of the correlation function and calculate the coherence ratio of the density matrix.These spectrums include constant spectrum,Lorentz spectrum,gaussian spectrum and 1/f spectrum.We find that for the noises with either a constant spectrum or the Gaussian spec-trum or the Lorentz spectrum,the coherence ratio decrease with time.Whereas,for the 1/f noise,the conclusion becomes more complicated.For the short-time limit,the coherence ratio decrease with square of time.But for long time limit the coher-ence ratio is approximated to a constant.With these results,we propose a method to suppress the decoherence of the density matrix.Additionally,we confirm these analytical results by numerical simulation.In the fifth chapter,we study a two-particle quantum-walk system with a non-abelian symmetry by coordinate Bethe ansatz method.We discuss the integrability of such model in each subspace for every irreducible representation.For each inte-grable subspace,we obtain the Bethe ansatz equation and verify the corresponding consistent conditions.Especially for two dimensional representation,the consistent condition is non-trivial.For un-integrable subspace we also give a explanation.In the last chapter,we summarize these thesis briefly.