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World Congress on Intelligent Control and Automation

Manipulation of NOT gate in Non-Markovian open quantum systems

Jianxiu Liu, Shuang Cong

Published Jun 1, 2014• 4 citations
Publisher

Abstract

The manipulation of gate operators in Non-Markovian open quantum systems are studied in this paper. A Lyapunov function based on operator distance is proposed. An additional adjustable item and the time-dependent control gain in Lyapunov control laws are used to improve control accuracy. In addition, another set of control laws is designed to maintain gate operators within given accuracy as long as possible. The proposed scheme is applied to prepare single qubit NOT gate in a two-level Non-Markovian system by the numerical simulations. The results demonstrate that the error can be limited within the level of 10−4for a significant long time and the system under the proposed control law is robust against the uncertainties of system parameters.

DOI
10.1109/wcica.2014.7052895

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The manipulation of gate operators in Non-Markovian open quantum systems are studied in this paper. A Lyapunov function based on operator distance is proposed. An additional adjustable item and the time-dependent control gain in Lyapunov control laws are used to improve control accuracy. In addition, another set of control laws is designed to maintain gate operators within given accuracy as long as possible. The proposed scheme is applied to prepare single qubit NOT gate in a two-level Non-Markovian system by the numerical simulations. The results demonstrate that the error can be limited within the level of 10−4for a significant long time and the system under the proposed control law is robust against the uncertainties of system parameters.
In this paper the control laws of preparing quantum gates are designed based on Lyapunov stability theorem for two level open quantum systems. We propose a novel Lyapunov function according to the matrix logarithm function, which has higher accuracy and faster convergence speed by comparing them with those of the Lyapunov function of distance. Based on the proposed function, we design two types of control laws to prepare quantum gates for different systems including Markovian quantum systems with phase damping and amplitude damping, non-Markovian quantum systems and closed quantum systems. Furthermore, the system robustness when the Hamiltonian contains uncertainty is further investigated. In order to verify the superiorities of proposed function and control method, NOT gates are prepared by the designed control laws for different systems in the numerical experiments, and the results are comparatively analyzed.
The weak-coupled two-level open quantum system described by non-Markovian time-convolution-less master equation is investigated in this paper. Based on the Lyapunov stability theorem, the system control laws used to transfer state are designed. The characteristics of the purity changes in the non-Markovian open quantum systems are studied. The effectiveness of the control system is ensured by adjusting the parameters of control laws. Three simulation experiments are implemented under the Matlab environment, they are the states transfer from the eigenstate to the superposition state, from the superposition state to the superposition state, and from the mixed state to the superposition state, respectively. Numerical experimental results show that the target state can be reached from different initial states under the action of the control laws designed, and demonstrate the effectiveness of quantum Lyapunov control method proposed.
We design the control laws which can be used to prepare operators based on quantum Lyapunov control method for two level Markovian open quantum systems in this paper. For the operator preparation, an often used Lyapunov function is the distance Vdisbetween the evolution operator U (t) and the target operator Uf. We construct a novel Lyapunov function V which is from the matrix logarithm function log U†fU (t). The convergence speeds and numerical values of V and Vdisare analyzed in detail. The control laws which are used to prepare operators are designed based on the proposed V. Numerical experiments are carried out to verify the superiority of the designed control laws.
A control method for trajectory tracking in non-Markovian open quantum systems is proposed in this paper. By inducing the error between the controlled state and the target state as the error state, the trajectory tracking problem of the quantum system is divided into two steps, which the regulation control of the error state and the keeping control of the error state. With the help of Lyapunov stability theorem, the tracking control law is designed to enable the non-Markovian open quantum system to track the evolutionary target quantum system over time. The control parameters are adjusted by simulation experiments, and the numerical simulation experiments are carried out. The tracking control of the quantum system when the initial states of the controlled system and the initial states of the target system are superposition states is implemented in simulation experiments. The experimental results demonstrate the feasibility of the proposed control method and the analysis is given.
One main problem of quantum control in quantum systems is the manipulation of states including state transfer and state tracking. We design control fields to realize arbitrary state transfer for open quantum system by Lyapunov stability theory, and investigate the state transfer for non-Markovian system with phase relaxation and energy dissipative relaxation. The numerical simulations illustrate that arbitrary state (eigenstate, superposition state or mixed state) transfer and maintenance for non-Markovian system can be realized under Lyapunov control function by an external steady control field of proper amplitude, with a success rate of more than 98%. When the right control field and function are implemented, not only the decoherence is compensated completely but also the purity of quantum states is maintained in the process of state transfer.
In this paper, the control laws based on the Lyapunov stability theorem are designed for a two-level open quantum system to prepare the Hadamard gate, which is an important basic gate for the quantum computers. First, the density matrix interested in quantum system is transferred to vector formation. Then, in order to obtain a controller with higher accuracy and faster convergence rate, a Lyapunov function based on the matrix logarithm function is designed. After that, a procedure for the controller design is derived based on the Lyapunov stability theorem. Finally, the numerical simulation experiments for an amplitude damping Markovian open quantum system are performed to prepare the desired quantum gate. The simulation results show that the preparation of Hadamard gate based on the proposed control laws can achieve the fidelity up to 0.9985 for the different coupling strengths.