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IEEE Transactions on Fuzzy Systems

Control Design of Uncertain Quantum Systems With Fuzzy Estimators

Chunlin Chen, Daoyi Dong, James Lam, Jian Chu, Tzyh‐Jong Tarn

Published Feb 3, 2012• 34 citations
Publisher

Abstract

An approach of control design using fuzzy estimators (FEs) is proposed for quantum systems with uncertainties. Two types of quantum control problems are considered: 1) control of a pure-state quantum system in the presence of uncertainties and 2) control design of quantum systems with initial mixed states and uncertainties. For the first type of tasks, a partial feedback control scheme with an FE is presented to design controllers. In this scheme, an FE is trained to estimate the quantum state for feedback control of a quantum system, and controlled projective measurement is used to assist in controlling the system. For the second type of quantum control tasks, a probabilistic fuzzy estimator (PFE) is trained to estimate the quantum state for control design of a quantum system with an initial mixed state, and a corresponding control algorithm is proposed to design a control law that drives the system from the mixed state to a target pure state. Two examples of two-spin-1/2systems are also presented and analyzed to demonstrate the process of control design and potential applications of the proposed approach.

DOI
10.1109/tfuzz.2012.2186817

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An approach of control design using fuzzy estimators (FEs) is proposed for quantum systems with uncertainties. Two types of quantum control problems are considered: 1) control of a pure-state quantum system in the presence of uncertainties and 2) control design of quantum systems with initial mixed states and uncertainties. For the first type of tasks, a partial feedback control scheme with an FE is presented to design controllers. In this scheme, an FE is trained to estimate the quantum state for feedback control of a quantum system, and controlled projective measurement is used to assist in controlling the system. For the second type of quantum control tasks, a probabilistic fuzzy estimator (PFE) is trained to estimate the quantum state for control design of a quantum system with an initial mixed state, and a corresponding control algorithm is proposed to design a control law that drives the system from the mixed state to a target pure state. Two examples of two-spin-1/2systems are also presented and analyzed to demonstrate the process of control design and potential applications of the proposed approach.
A partial feedback control scheme with a probabilistic fuzzy estimator (PFE) is presented for the robust control of quantum systems. In this scheme, a probabilistic fuzzy simulator is trained to estimate the quantum states for feedback control of quantum systems. Usually, the estimated state is fed back to design a controller. However, when the estimated quantum state is an almost-eigenstate, a projective measurement will be triggered for the quantum system and the measurement results will be fed back to construct the controller and regulate the fuzzy estimator. This scheme is a partial feedback strategy with controlled discontinuous measurement, where the quantum measurement serves as a control tool and is helpful for driving the quantum system to a desired state tracking even in the presence of unknown disturbances and stochastic noises. An example of a two-spin-¿ system is also presented to demonstrate the proposed approach.
A hybrid control approach for uncertain quantum systems is proposed using probabilistic fuzzy estimators (PFE) and quantum reinforcement learning (QRL). This hybrid control design involves coherent control with PFE and learning control via QRL. The problems of controlling a quantum system from an initial state to a pointed target state are studied in this paper, where we assume that the initial quantum state is a mixed state and the target quantum state is a controllable pure state within a wavefunction controllable subspace. First, the initial quantum system is controlled coherently with the help of a PFE. When the controlled system is estimated to be likely to collapse to an expected eigen state, trigger the measurement and the quantum system collapses to an eigen state in the wavefuntion controllable subspace with a high probability. Then the quantum system is driven to the target state with admissible controls, where the control sequence is learned and optimized with QRL. An example is presented and analyzed to demonstrate the control process.
The design of optimal final-state controllers of quantum-mechanical systems that are insensitive to errors in the molecular Hamiltonian or to errors in the initial state of the system is considered. Control arises through the interaction of the system with an external field; the goal is optimal design of these latter fields for various physical objectives in the presence of system uncertainty. Sensitivity to modeling errors and other uncertainties in the molecular Hamiltonian is minimized by considering averaged costs for a family of Hamiltonian functions H0{\mathit{H}}_{0}H0​(\ensuremath{\alpha}) indexed by the random variable \ensuremath{\alpha} taking values on a compact set in Euclidean space. Similarly, sensitivity of the optimal control to the initial state is minimized by viewing the initial condition as a Hilbert-space-valued random variable and considering an optimization problem with a cost functional that is averaged over the class of initial conditions. A precise formulation of the control problem is given, and its well-posedness is established. Cost propagators are defined to display the dependence of the performance index on the initial conditions explicitly, which allows analytic averaging of initial conditions.The constrained optimization problem is reduced to an unconstrained optimization problem by the introduction of Lagrange-multiplier operators. Necessary conditions for the unconstrained problem provide the basis for a gradient search for an optimal solution. Finite-difference schemes are utilized to provide a numerical approximation of the optimal control problem. Numerical examples are given for final-state control of a diatomic molecule represented by a Morse potential illustrating design for systems with initial-phase uncertainty and parametric uncertainty. The resultant insensitive controllers execute different strategies depending on the design requirements. The controller designed to be insensitive to errors in the initial phases adopts a strategy of phase imprinting during the initial stages of the control interval to compensate for a lack of knowledge of the initial phases. It is also shown that it is not possible to coerce a system from a state with completely random initial phases to a correlated state using the class of averaged controllers considered here. The controller designed to be insensitive to parametric Hamiltonian errors adopts a strategy of amplitude restraint to prevent the wave packet from taking significant excursions into the regions where the potential is uncertain. The interesting structure exhibited by the controllers in response to the different design requirements, and the superior performance of the insensitive controllers when compared with controllers designed at nominal phases and parameters, illustrate the usefulness of the cost-averaging technique for design in the presence of uncertainties.
Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with uncertainties. The SLC method includes two steps of training and testing. In the training step, an augmented system is constructed using artificial samples generated by sampling uncertainty parameters according to a given distribution. A gradient flow-based learning algorithm is developed to find the control for the augmented system. In the process of testing, a number of additional samples are tested to evaluate the control performance, where these samples are obtained through sampling the uncertainty parameters according to a possible distribution. The SLC method is applied to three significant examples of quantum robust control, including state preparation in a three-level quantum system, robust entanglement generation in a two-qubit superconducting circuit, and quantum entanglement control in a two-atom system interacting with a quantized field in a cavity. Numerical results demonstrate the effectiveness of the SLC approach even when uncertainties are quite large, and show its potential for robust control design of quantum systems.
Robust control design for quantum systems has been recognized as a key task in the development of practical quantum technology. In this paper, we present a systematic numerical methodology of sampling-based learning control (SLC) for control design of quantum systems with Hamiltonian uncertainties. The SLC method includes two steps of “training” and “testing and evaluation”. In the training step, an augmented system is constructed by sampling uncertainties according to possible distributions of uncertainty parameters. A gradient flow based learning and optimization algorithm is adopted to find the control for the augmented system. In the process of testing and evaluation, a number of samples obtained through sampling the uncertainties are tested to evaluate the control performance. Numerical results demonstrate the success of the SLC approach. The SLC method has potential applications for robust control design of quantum systems.
This paper proposes a new robust control method for quantum systems with uncertainties involving sliding mode control (SMC). SMC is a widely used approach in classical control theory and industrial applications. We show that SMC is also a useful method for robust control of quantum systems. In this paper, we define two specific classes of sliding modes (i.e. eigenstates and state subspaces) and propose two novel methods combining unitary control and periodic projective measurements for the design of quantum SMC systems. Two examples including a two-level system and a three-level system are presented to demonstrate the proposed SMC method. One of the main features of the proposed method is that the designed control laws can guarantee the desired control performance in the presence of uncertainties in the system Hamiltonian. This SMC approach provides a useful control theoretic tool for robust quantum information processing with uncertainties.