Parameter Estimation In Typical Quantum Systems And Quantum State Tomography Algorithm

Quantum parameter estimation,which studies how to estimate parameters in quantum systems,is important in scientific basic field and plays an guiding role in practical application.Quantum tomography,the essential technology to know about the quantum state generation source,studies how to select measurement and construction algorithm to infer the quantum state in a quantum system.In this thesis,we study the parameter estimation of unknown parameters and unknown state in quantum systems.The main content is shown as follows:Firstly,on condition that the measurement operators are restricted,we use a two-step scheme to estimate the parameter in a one-qubit quantum system.In terms of the Fisher information(FI)of the intermediate variable containing the interested parameter,we determine the optimal parameters in the measurement operators and the initial state.Basing on the optimal setup,we can estimate the interested parameter and get the optimal evolution time in terms of the FI of the parameter,on condition that a priori information of the parameter is given.The system optimal set-up and estimation precision of the parameter to infer considering Fisher information are identical to the ones in terms of quantum Fisher information in previous research.Secondly,we use periodical projective measurement to estimate the dephasing parameter and the tunneling parameter from a spin-boson system.There are two benefits to use periodical projective measurement:first,it is easy to be implemented;second,the state prepared by measurements in the previous period is the initial state of the current period so that the quantum state is recycled.The dephasing parameterγcan be always estimated when projective measurement bases are chosen asθ=π/2,φ=0.Based on the estimated value ofγand the interval information oft he tunneling parameterΔ,we can select another measurement bases(θ=π/4 andφ=π/2)to obtain the estimated value ofΔ.A coherent control is indispensable to estimateΔifγis in the interval ofΔ;whereas the control is not necessary ifγis out of the known interval ofΔ.We establish the relation between the optimal period time and the parameterγorΔin terms of Fisher information.Thirdly,we analyze the influence of the uncertainties of periodical projective measurements on the precision of the estimated dephasing and tunneling parameters in a spinboson model with dephasing noise.FI of dephasing parameter and tunneling parameter is the figure of merit for the precision of parameter estimation.We get the relation between the FI lower bound of estimated parameters and the uncertainties of measurements.The results are useful for determining the domain in which the uncertainties of measurements have to stay to guarantee the lower bound of parameter estimation precision in a parameter estimation scheme.Fourth,we study how to use quantum algorithms to realize the process of quantum state reconstruction.Based on the process of linear regression reconstruction algorithm,different quantum algorithms are used through the process so that a whole quantum algorithm is formed to deal with the quantum state reconstruction.When both the condition number of the matrix and the desired precision satisfy the condition of using the quantum algorithm to accelerate,as well as the necessary number of quantum states is O(d),the entire quantum algorithm can reduce the time complexity of the quantum state tomography process describing by linear regression model from O(d~4)to O(dpoly log d).