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IEEE/CAA Journal of Automatica Sinica

Preparation of Hadamard Gate for Open Quantum Systems by the Lyapunov Control Method

Nourallah Ghaeminezhad, Shuang Cong

Published Apr 5, 2018• 25 citations
Publisher

Abstract

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.

DOI
10.1109/jas.2018.7511084

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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.
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.
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.
This paper studies the realization of the quantum Hadamard gate of spin with a single electron. First the canonical form of the unitary rotation gate which is feasibly applied in plane control fields is given. Because the Hadamard gate can't be realized directly by applying the plane control fields, so the matrix operator of Hadamard gate is decomposed into two matrices which are feasible in canonical forms. These two canonical matrices correspond to the rotate-operation and reflect-operation of the vector coordinates on the Bloch sphere, respectively. Then the Hadamard gate can be realized in two steps by choosing appropriate control fields, in such a way the realization of Hadamard gate becomes to design external tailored control fields. Finally, the control fields are designed based on Lyapunov method. Numerical simulation experiments and results analysis are given.
The unitary matrix was constructed by using eigenstates of the controlled quantum system so as to perform a coordinate rotation on the controlled system.In such a way the local phases of all states of the system were eliminated.Then the error between the controlled state and the target state after transformation was selected as the Lyapunov function,and the system control law was designed based on Lyapunov stability theorem with the pre-condition that the system is kept stable.The detailed design procedure of the controller needed for preparation of the system states and the comparison analysis of the results were given.Finally,the system simulation experiments on a spin-1/2 particle system were implemented to prepare both the eigen and superposition target states with the proposed control method,and the relation between the system state evolution time and control value was analyzed with different parameters.
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.