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IEEE Conference on Decision and Control (CDC)

Lyapunov-based control of quantum systems

Symeon Grivopoulos, Bassam A. Bamieh

Published Jul 8, 2004• 91 citations
Publisher

Abstract

We propose feedback control laws for quantum systems, based on Lyapunov design. These feedback laws locally asymptotically stabilize any desired eigenstate of the system Hamiltonian. The target eigenstate and the size of its region of attraction can be tailored by the choice of design parameters. These feedback control laws are used to design open loop controls for state transfer problems. We illustrate the method with a transfer problem between bound states of a molecular Morse oscillator.

DOI
10.1109/cdc.2003.1272601

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We propose feedback control laws for quantum systems, based on Lyapunov design. These feedback laws locally asymptotically stabilize any desired eigenstate of the system Hamiltonian. The target eigenstate and the size of its region of attraction can be tailored by the choice of design parameters. These feedback control laws are used to design open loop controls for state transfer problems. We illustrate the method with a transfer problem between bound states of a molecular Morse oscillator.
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.
In the design of complex quantum systems like ion traps for quantum computing, it is usually desired to stabilize a particular system state or make the system state track a desired trajectory. Several control theoretical approaches based on feedback seem attractive to solve such problems. But the uncertain dynamics introduced by measurement on quantum systems makes the synthesis of feedback control laws very complicated. Although we have not explicitly modeled the change in system dynamics due to measurement (we have assumed weak measurements), this is a first step towards a more detailed analysis and closed-loop feedback design. Here, we present a Lyapunov-based control approach on the lines of that developed by Mirrahimi, Rouchon, Turnici (2005). The states are assumed to be obtained from weak measurements. The Lyapunov control technique has not been applied to realistic quantum systems so far. We have extended and applied the technique to two realistic physical systems - the quantum harmonic oscillator and the n-qubit system. We also propose to extend this concept to ion traps.
The paper proposes a gradient method for feedback control and stabilization of quantum systems using Schröodinger’s and Lindblad’s descriptions. The eigenstates of the quantum system are defined by the spin model. First, a gradient-based control law is computed using Schröodinger’s description. Next, an estimate of state of the quantum system is obtained using Lindblad’s differential equation. In the latter case, by applying Lyapunov’s stability theory and LaSalle’s invariance principle one can compute a gradient control law which assures that the quantum system’s state will track the desirable state within acceptable accuracy levels. The performance of the control loop is studied through simulation experiments for the case of a two-qubit quantum system.
Recently, feedback-based quantum algorithms have been introduced to calculate the ground states of Hamiltonians, inspired by quantum Lyapunov control theory. This paper aims to generalize these algorithms to the problem of calculating an eigenstate of a given Hamiltonian, assuming that the lower energy eigenstates are known. To this aim, we propose a new design methodology that combines the layer wise construction of the quantum circuit in feedback-based quantum algorithms with a new feedback law based on a new Lyapunov function to assign the quantum circuit parameters. We present two approaches for evaluating the circuit parameters: one based on the expectation and overlap estimation of the terms in the feedback law and another based on the gradient of the Lyapunov function. We demonstrate the algorithm through an illustrative example and through an application in quantum chemistry. To assess its performance, we conduct numerical simulations and execution on IBM's superconducting quantum computer.
In this paper, a Lyapunov-based feedback control (LFC) is proposed for the state transfer of a two-level stochastic quantum system. The quantum state is described in the Cartesian coordinate system. An exponential function is selected as the Lyapunov function to ensure the stability of the system. The Lyapunov-based feedback control strategy is designed according to the stochastic Lyapunov stability theorem. Numerical simulations results show that the proposed LFC strategy has the ability to transfer the stochastic system from any initial mixed state to the desired eigenstate, and the LFC has high convergence rate and high state transfer fidelity.
Abstract State transition control of open quantum systems is a research hotspot and difficulty. The control method in the light of the Lyapunov stability theory is a common method. Current research shows that if the open quantum systems meet the constraints that the free Hamiltonian is strong regular and the interaction Hamiltonian is fully connected, by devising appropriate control fields in the light of the Lyapunov stability theory, the open quantum system can move to the desired final state. In view of the fact that most practical open quantum systems do not meet these two constraints, this paper propose a control method. The implicit function perturbation terms are introduced into the control fields to resolve the problem that the open quantum systems do not meet these two convergence constraints. And the main control fields are devised in the light of the Lyapunov stability theory to make the state transition of the open quantum systems converging to the desired final state.