Research On Dynamical Robustness Of Two Dimensional Topological Quantum Memory

Quantum computer is one possible new paradigm which can break through the upcoming limit ofclassical computer.Qbit is the elemental unit ofquantum computation as the bit of classical computation.Any quantum system inevitably interacts with environment and no system will ever be free from week perturba-tion.The errors caused by the thermal effect or imperfection are what we have to face to.The quantum error-correcting code is invented to solve the problem as what the classical error-correcting code does in classical information theory.The process of quantum error-correction is keeping detecting the incident errors and then correcting them to make information resilient for long time,which is called active quantum error-correction.However,as all we knowthe information stored in the magnetic disk needs not to be corrected constantlyThe ferromagnetic order protects the classical information so well that the information can be stored intactly for decades,as long as the environment is at room temperature and with-out strong magnetic field.People want to find the same robust feature ofpassive quantum error-correction.One possible scenario is exploiting topological order to protect quantum information.Toric code invented by Kitaev in 1997 is the simplest topological ordered model,whose ground state space is robust against any local perturbation.The ground space can be used to code two quantum bit-s that can be a quantum memory.The quasi-particle excitations oftopological ordered system present exotic statistics,which can realize fault-tolerant quantum computation.This new research field is called topological quantum computation.Topological systems possess a long-range pattern of entanglement.The entangle-ment entropy of their ground state wave function conf'orms the boundary law,as any gapped local Hamiltonian does.In addition to the boundary dependent ter-m,one constant term appears in the topological ordered wave function,which is called topological entanglement entropy.It serves as the order parameter of these nontrivial phases.Two dimensional toric code is robust in the sense that pertur-bations in the Hamiltonian will not change the topological nature of the ground state wave function.However,in order to exploit it for application such as passive error-correcting quantum memory and quantum computation,these states need to be also robust both at finite temperature and dynamically.The analytical calculation shows that,at finite temperature,topological entropy remains only the half of the value at zero temperature when the gauge structure is preserved,while completely vanishes when the gauge structure is broken.More theoretic re-searches conclude that any two dimensional local commuting Hamiltonian is not topological ordered at finite temperature.In this dissertation,we calculate the topological entropy of two dimensional toric code after time evolution at zero temperature.The protocol ofthe evolution is quantum quench.We first prepare the initial state as the toric code ground state,then the external fields are turned on suddenly.The initial state will then evolve by the time evolution operator generated by the new Hamiltonian.We obtain a analytical formula ofpurity for the subsystem and also the method of calculating both 2-Renyi entropy and topological 2-Renyi entropy.We find that if the quench Hamiltonian possesses local gauge symmetry,the topological entropy will evolve to the half of the initial value:If the quench Hamiltonian has no local gauge symmetry,the topological entropy will vanish completely.This result implies that the long-time behavior of'topological entropy is thermal.It means that the toric code is not a robust quantum memory both at finite temperature and af ter a quantum quench.The inf ormation of the initial state will completely vanish if the system is thermalized.But localization inducetopological d by disorder can preserve the initial infor-mation partially.We calculate the Renyi entropy with disorder,and find that we can make it resilient by increasing the disorder strength.Neverthe-less,the above-mentioned method is applied to only small subsystem size,so we need to consider bigger subsystem to obtain stronger evidences.Next we calculate the expectation value of Wilson loop and entanglement entropy with large subsys-tem size.We find that these two quantities keep boundary law when the disorder is added to the system,which implies that the dynamical localization keeps the topological phase resilient.This result shows that dynamical localization protects the topological quantum memory after a quantum quench at zero temperature.