Quantum Walk On A Complex Graph And Quantum Gates For A Bose-Einstein Condensate System

We analyzed the continuous-time quantum walk on the Chimera diagram,which is the topological diagram of the underlying physical structure of the D-Wave quantum annealing computer.We have discovered unique quantum walk features that are starkly distinguishes classical quantum walk,such as localization.In order to further study the impact of different underlying structure changes on quantum annealing computer,we studied the enhanced and weakened variants of the Chimera graph.The enhanced variant is achieved by increasing the coupling between qubits,and correspondingly,the weakened variant is achieved by cutting off the coupling between certain qubits.Reflected in the structure of the Chimera graph,it means to increase or decrease the connectivity of the graph.We construct a generating set for a suitable subgroup of graph isomorphisms to the Chimera graph to explain the results of quantum walk.Each element in the generating set corresponds to a symmetry operation,and the symmetry operation is commutative with the Hamiltonian of the quantum walk.The quantum walk Hamiltonian and these symmetry operators together provide a complete set of commuting observables,which can be used to completely distinguish the energy levels and existing degenerate states of quantum walk system.Our quantum walk characterization of the Chimera graph and its variants yields valuable insights into graphs used for designing quantum-annealers.In the quantum walk problem on the Chimera graph,we mainly have the following innovations:1)By analyzing the symmetry of the graph,we analytically obtain the eigenvalues and eigenvalues of the quantum walk Hamiltonian on large-scale graphs;2)we study the Chimera structure and its variants,which have reference value for the underlying hardware topology of next-generation quantum annealing calculators;3)our symmetry-based method can be used to analyze the evolution and dynamic behavior of quantum walk on other large-scale graphs.Another work of this thesis studies the realization of fast and high-fidelity quantum gates on nonlinear quantum systems.Through the quantum control strategy,we have realized the quantum gate on the motional states of the Bose-Einstein condensate system,which overcomes the limitation brought by the inherent nonlinear evolution of the system.The dynamic evolution of the Bose-Einstein condensate system can be described by the Gross-Pitaevskii equation,which is a nonlinear Schrodinger equation.Due to the existence of nonlinearity,the principle of linear superposition is no longer hold.This brings huge challenges to the realization of quantum gates.In fact,in nonlinear systems,quantum gates need to be redefined.We put the quantum optimal control problem in the framework of the optimization problem,and then use standard global optimization software to solve it.We propose an effective strategy to achieve a Hadamard gate with an average fidelity of 99%and a gate time of 1ms.We used this Hadamard gate to demonstrate Ramsey interference and achieved a contrast ratio of 99.97%.We give the definition and realization of quantum gate in nonlinear system,which provides a powerful tool for studying quantum phenomena of nonlinear system.In the problem of realizing quantum gates on Bose-Einstein condensate systems,we mainly have the following innovations:1)Our work includes a definition of nonlinear quantum gates,which combines the salient features of linear quantum gates but connects with nonlinear quantum mechanics,for which nonlinear unitary evolution only preserves the norm but not the inner product;2)our control method needs to sample many initial states;this is because the dynamics of Bose-Einstein condensate systems are nonlinear,which violates the superposition rule of linear quantum mechanics;3)we search for the optimal control parameters of the external potential field in the frequency domain instead of the time domain,which will effectively reduce the size of the search space;4)we devise a fast high-fidelity nonlinear Hadamard gate and use this gate to demonstrate a high-contrast nonlinear Ramsey interferometer;5)out strategy can also implement other types of quantum gates in many-body quantum systems.