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arXiv:1203.5373 · Physical Review A

Optimal Lyapunov-based quantum control for quantum systems

S. C. Hou, Muhammad Aslam Khan, X. X. Yi, Daoyi Dong, Ian R. Petersen

Published Aug 21, 2012• 118 citations
arXivHTMLPDF

Abstract

Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as $t\ensuremath{\rightarrow}\ensuremath{\infty}$, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.

DOI
10.1103/physreva.86.022321

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Quantum Lyapunov control was developed in order to transform a quantum system from arbitrary initial states to a target state. The idea is to find control fields that steer the Lyapunov function to zero as t\ensuremath→\ensuremath∞t\ensuremath{\rightarrow}\ensuremath{\infty}t\ensuremath→\ensuremath∞, meanwhile the quantum system is driven to the target state. In order to shorten the time required to reach the target state, we propose two designs to optimize Lyapunov control in this paper. The first design makes the Lyapunov function decrease as fast as possible with a constraint on the total power of control fields, and the second design has the same purpose but with a constraint on each control field. Examples of a three-level system demonstrate that the evolution time for Lyapunov control can be significantly shortened, especially when high control fidelity is required. Besides, this optimal Lyapunov-based quantum control is robust against uncertainties in the free Hamiltonian and decoherence in the system compared to conventional Lyapunov control. We apply our optimal design to cool a nanomechanical resonator where a shorter cooling time is found with respect to the cooling time by the conventional Lyapunov design.
Taking a two-level system as an example, we show that a strong control field may enhance the efficiency of optimal Lyapunov quantum control in [Hou et al., Phys. Rev. A \textbf{86}, 022321 (2012)] but could decrease its control fidelity. A relationship between the strength of the control field and the control fidelity is established. An extended technique, which combines free evolution and external control, is proposed to improve the control fidelity. We analytically demonstrate that the extended technique can be used to design a control law for steering a two-level system exactly to the target state. In such a way, the convergence of the extended optimal Lyapunov quantum control can be guaranteed.
Quantum control is responsible for many key problems in quantum information processing, such as decoherence suppression, quantum state engineering and quantum gate, etc. In recent years quantum control has become a major topic in the fields of quantum optics and quantum information. Fast control of the state of quantum system is effective for both decoherence suppression caused by the environment and information processing in quantum state transfer and tracking. Comparing the standard Lyapunov control function with bang-bang Lyapunov control function, we design an optimized quantum Lyapunov control function, and analyse the influence of control laws on population transfer probability and control time contrastively. The simulation results indicate that the optimized quantum Lyapunov control strategy effectively shortens the control time and avoids high-frequency oscillation that occurs in bang-bang Lyapunov control, therefore, strengthens the robustness against external interference.
By using a quadratic Lyapunov function with un-known parameters and a particle swarm optimization (PSO) algorithm, this paper proposes a new Lyapunov control scheme for quantum systems. This approach can achieve high-probability population transfer to a given target state. For the case when the target state is an eigenstate of the internal Hamiltonian, we introduce a virtual control law into the system model and design corresponding control law. The stability of the system under the action of the Lyapunov control law is analyzed via the LaSalle invariance principle. For the case when the target state is a superposition state, we design a control law by performing a unitary transformation for the quantum system model under consideration. To achieve desired state transfer, we further introduce a PSO algorithm to search for the unknown parameters contained in the control law. Numerical results on a five-level quantum system and a three-qubit system are presented to demonstrate the effectiveness of the proposed approaches.
A scheme to drive and manipulate a finite-dimensional quantum system in decoherence-free subspaces (DFSs) by Lyapunov control is proposed. Control fields are established by the Lyapunov function. This proposal can drive the open quantum system into DFSs and manipulate it to any desired eigenstate of the free Hamiltonian. An example which consists of a four-level system with three long-lived states driven by two lasers is presented to demonstrate the scheme. We have performed numerical simulations of the dynamics of the four-level system, which show that the scheme works well.
A Lyapunov-based control scheme is presented to drive closed quantum systems into any target eigenstate with as high population as possible by the quantum-behaved particle swarm optimization (PSO) algorithm. Based on a Lyapunov function with a Hermitian operator to be constructed, a control law with the unknown parameters contained in the Hermitian operator is designed. To achieve high-population state transfer to the target state, we first initialize those unknown parameters by choosing a path to the target state in its energy-level connectivity graph and setting their values along the path. Then, a set of optimal parameters is found by the quantum-behaved PSO algorithm. Finally, numerical simulation experiments are performed on a five-level quantum system and a four-qubit system to demonstrate the effectiveness of the control scheme in this paper.
A Lyapunov-based method is presented for stabilizing and controlling of closed quantum systems. The proposed method is constructed upon a novel quantum Lyapunov function of the system state trajectory tracking error. A positive-definite operator in the Lyapunov function provides additional degrees of freedom for the designer. The stabilization process is analyzed regarding two distinct cases for this operator in terms of its vanishing or non-vanishing commutation with the Hamiltonian operator of the undriven quantum system. To cope with the global phase invariance of quantum states as a result of the quantum projective measurement postulate, equivalence classes of quantum states are defined and used in the proposed Lyapunov-based analysis and design. Results show significant improvement in both the set of stabilizable quantum systems and their invariant sets of state trajectories generated by designed control signals. The proposed method can potentially be applied for high-fidelity quantum control purposes in quantum computing frameworks.