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IEEE Journal of Selected Topics in Quantum Electronics

Entanglement Creations and Quantum Gate Implementations of Spin Qubits With Lyapunov Control

Yi‐Hao Kang, Yan Xia

Published Dec 13, 2019• 1 citation
Publisher

Abstract

In this paper, we propose a scheme for the entanglement creations and the quantum gate implementations of spin qubits via Lyapunov control. We first show that three types of Bell states can be generated in one step with the traditional Lyapunov control. After that, the traditional Lyapunov control is generalized with the design of two Lyapunov functions that characterize the distance between two unitary operators. Thus, the Lyapunov control can be also used in constructing desired unitary operations. The performance of the entanglement creations and the quantum gate implementations are both estimated via numerical simulations. The results show the scheme holds robustness against the errors of the fields and couplings. As the generalization of Lyapunov control can not only be used in spin systems, the scheme may benefit the quantum information processing in different quantum systems.

DOI
10.1109/jstqe.2019.2958924

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In this paper, we propose a scheme for the entanglement creations and the quantum gate implementations of spin qubits via Lyapunov control. We first show that three types of Bell states can be generated in one step with the traditional Lyapunov control. After that, the traditional Lyapunov control is generalized with the design of two Lyapunov functions that characterize the distance between two unitary operators. Thus, the Lyapunov control can be also used in constructing desired unitary operations. The performance of the entanglement creations and the quantum gate implementations are both estimated via numerical simulations. The results show the scheme holds robustness against the errors of the fields and couplings. As the generalization of Lyapunov control can not only be used in spin systems, the scheme may benefit the quantum information processing in different quantum systems.
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.
Quantum control concerns driving quantum systems to the desired properties or states. This objective is hindered by the non-deterministic nature of quantum systems and the symmetric topology of the quantum state space. This paper introduces a coupling Lyapunov functions approach to counter for the stochastic nature of quantum measurements and break the symmetric topology of resulting filter state space. As an example, we then apply it to synthesize non-smooth control almost surely globally stabilizing the desired Bell state-the maximally entangled two-qubit state with various applications. It turns out that this approach enables the short time convergence of the system state, which is essential in the control of entanglement. More interestingly, its combination with the generalized concept of SWM (simultaneous weak measurement)-induced quantum state reduction allows us to deterministically generate the maximally entangled Bell state with little non-local effects. We use numerical illustration to validate the proposed scheme.
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 review the scheme of quantum Lyapunov control and its applications into quantum systems. After a brief review on the general method of quantum Lyapunov control in closed and open quantum systems, we apply it into controlling quantum states and quantum operations. The control of a spin-1/2 quantum system, driving an open quantum system into its decoherence free subspace (DFS), constructing single qubit and two-qubit logic gates are taken to illustrate the scheme. The optimalization of the Lyapunov control is also reviewed in this article.