Applicability Of Margolus-Levitin Bound Of Quantum Speed Limit

Quantum speed limit(QSL)originates from Heisenberg uncertainty principle.In 1945,Mandelstam and Tamm interpreted the uncertainty relationship between time and energy as a description which was time scale of quantum unitary dynamics on basis of previous theories.They proposed a quantum speed limit bound——MT bound,which was related to the standard deviation of system energy.In 1998,Margolus and Levitin proposed another quantum speed limit bound which was called ML bound.ML bound related to the average energy of the system.With the above two bounds as the mainstream,quantum speed limit had developed vigorously and attracted attention of many researchers.Quantum speed limit answers the fundamental question of whether there is a lower bound on the evolution time between different quantum states,which has profound physical meaning.The study of quantum speed limit helps to further improve the framework of quantum theory,and the corresponding results can be applied to many fields of physics,such as:guiding the fastest operation of quantum gates in quantum computing;deriving the maximum entropy yield in quantum thermodynamics;estimating the cosmic computing power in astrophysics,etc.In the process of extending quantum speed limit,physical meaning of MT bound is clear,so it is successfully extended to time-dependent systems and open systems.However,the generalization of the ML bound is more difficult,and the relevant generalizations proposed so far have certain shortcomings in terms of theoretical basis and application scope.This thesis expounds the geometric structure of Hilbert space through fiber bundle theory,and selects Fubini-study metric as the method to measure distance between two different points in this space.Hilbert space and Fubini-study metric combine to form a Riemannian manifold.Then,quantum speed limit is studied by using relevant knowledge of Riemannian manifold.Different from mathematical derivation of ML bound,the new view of ML bound from perspective of geometric properties is helpful to deepen the understanding of ML bound,and it is easier to reflect the scope of application and extension requirements of ML bound.In addition,the physical images of ML bound are clearer when applied to specific models.This thesis is divided into five chapters and organized as follows:The first chapter introduces development process and some achievements of quantum speed limit,also points out the current research hotspots and problems to be solved.The second chapter shows theoretical methods and basic knowledge needed in this thesis.Firstly,the derivation of MT bound and ML bound is introduced.Secondly,relevant knowledge of Riemannian manifolds,such as fiber bundle theory,evolution path length,geodesic equation and so on,is introduced,and derivation of geometric phase bound is briefly mentioned.Finally,types and functions of single qubit quantum gates are introduced.The third chapter firstly explores the initial state and Hamiltonian conditions required when quantum speed limit of two-level system satisfies geodesic evolution,giving corresponding proof and visualizing it in Bloch sphere.Then,theoretical deficiency of ML bound is filled——ML bound requires ground state energy of system Hamiltonian to be E0=0 without giving physical explanation.This thesis gives a reasonable explanation from two aspects:evolution speed of quantum states and "path-phase" relationship derived from Lie algebra,which provides a new theoretical support for ML bound and deepens the understanding of it.Finally,an expression of adjusting ground state energy to 0 by controlling phase is provided.The close relationship between ML bound and dynamic phase of quantum state evolution accumulation is proved.This phase expression is extended to the case of geometric phase.The fourth chapter establishes a spin-1/2 particle model of rotating magnetic field,analytical solutions of relevant parameters are calculated,and the evolution image is expressed in Bloch sphere.Considering the application of quantum speed limit in this model to construct single qubit quantum NOT gate and quantum phase-shift gate,which are discussed in adiabatic system and non-adiabatic system respectively.In adiabatic system,correctness of the evolution condition along the geodesic proposed in Chapter 3 is verified,and another restriction condition of ML bound is obtained by observing the case without geodesic evolution.The modulation mechanism of parameter pair evolution in Hamiltonian is qualitatively explored in non-adiabatic system,and two extreme cases of possible construction of quantum gate are predicted.The fifth chapter summarizes main conclusions of this thesis and takes the outlook on the future investigation.