Theoretical Study On Geometric Quantum Operations

Accurate manipulation of quantum states is the basic prerequisite for the realization of quantum information processing.However,there are two main challenges for the realization of accurate quantum operations.On the one hand,due to the existence of control errors,the control parameters in the Hamiltonian deviate from the ideal situation,which leads to the actual evolution of the quantum system deviating from the ideal evolution.On the other hand,practical quantum systems inevitably interact with the environment and this interaction leads to decoherence.Overcoming control errors and decoherence to achieve high fidelity quantum operations is the key point to the application of quantum information.In this thesis,we focus on the quantum operation approaches based on the geometric properties of quantum systems.First,we study the quantum operation approach based on geometric phases.Geometric phases are only dependent on evolution paths of quantum systems but independent of the details of the evolution process.Therefore,the quantum gates based on geometric phases can resist control errors.Second,we study the quantum operation approach based on geometric space curves.This approach exploits the geometric structure underlying the Schr?dinger equation to transform the problem of designing control pulse into the problem of finding the space curves that meet some requirements.It is an effective approach to realize accurate quantum operations.Finally,we study the problem of optimizing the quantum operation schemes by using the geometric properties of quantum systems.It is of great significance for the application of quantum information to reduce the energetic cost required to implement the quantum operations on the premise of ensuring the accuracy.Specifically,the following are the main results of this thesis:First,we put forward an approach to the realization of nonadiabatic geometric quantum gates.Based on a set of parameterized orthonormal bases,we obtain the general form of the Hamiltonian for nonadiabatic geometric gates by using the inverse engineering approach.By using the general form of the Hamiltonian,a universal set of nonadiabatic geometric gates can be realized with any desired evolution paths.Therefore,our approach makes it possible to realize geometric quantum gates with an economical evolution time so the influence of environment noises on the quantum gates can be minimized.Second,we propose a scheme to realize coherence-protected nonadiabatic geometric quantum computation.Based on the general form of the Hamiltonian for nonadiabatic geometric gates,we construct a driving Hamiltonian which can also realize the continuous dynamical decoupling.By using this Hamiltonian to drive the quantum system,not only the geometric feature of the nonadiabatic geometric gates is preserved,but also the system’s coherence is protected.Therefore,the geometric quantum computation scheme based on coherence-protected nonadiabatic geometric gates is not only robust to control errors but also resistant to environment-induced decoherence.Third,we propose a fast and robust scheme for population transfer between spin states which combines invariant-based inverse engineering and the quantum operation approach based on geometric space curves.Our scheme can not only be implemented quickly,but also effectively suppress the dominant noise in spin systems.These two advantages guarantee the accuracy of the population transfer between spin states.Moreover,the control parameters of the driving Hamiltonian in our scheme are easy to design because they correspond to the curvature and torsion of a three-dimensional visual space curve derived by using the quantum operation approach based on geometric space curves.Fourth,we establish the connection between the minimal value of the total energetic cost in the transitionless quantum driving process and the geometric property of the evolution path.It is found that the minimal value of the total energetic cost in the transitionless quantum driving process is equal to the length of the evolution path of the quantum system on the Riemannian manifold spanned by the control parameters,but independent of the evolution details such as the changing rate of the control parameters.Based on this geometric property,we further show that the optimal transitionless quantum driving with the minimum total energetic cost corresponds to the geodetic path on the Riemannian manifold,which provides an efficient method for optimizing the transitionless quantum driving.