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World Congress on Intelligent Control and Automation

Optimal control of quantum systems under different manipulation conditions

Jiahua Wei, Weiwei Zhou, Hong‐Yi Dai, Fengxing Zou, Ming Zhang

Published Jul 1, 2012• 1 citation
Publisher

Abstract

In this paper, we explicitly explore how to manipulate single qubits by one-rotation controls under various manipulation conditions. It is revealed that one can construct control Hamiltonian and adjust the control to manipulate quantum states. Furthermore, we comprehensively discuss how to optimize control magnitude in terms of a new kind of weighted time-energy performance. A comparison has been made among the optimal performances under different manipulation conditions. Three concrete examples indicate the feasibility and efficiency of this approach on optimal control of two-level quantum systems.

DOI
10.1109/wcica.2012.6358221

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In this paper, we explicitly explore how to manipulate single qubits by one-rotation controls under various manipulation conditions. It is revealed that one can construct control Hamiltonian and adjust the control to manipulate quantum states. Furthermore, we comprehensively discuss how to optimize control magnitude in terms of a new kind of weighted time-energy performance. A comparison has been made among the optimal performances under different manipulation conditions. Three concrete examples indicate the feasibility and efficiency of this approach on optimal control of two-level quantum systems.
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We consider a two-level quantum system (quantum bit), S, which can only be controlled by modulating the difference between the two energy levels. We propose a control scheme that utilizes an additional quantum bit, A, as an auxiliary system, and obtains manipulation of S through the interaction with A. For this scheme, we derive the minimum time control for any unitary transformation on the target system S, and the form of the optimal control law. We find that the optimal time protocol involves a preliminary preparation of the state of A, the control of S using interaction with A, and no simultaneous control of S and A. The optimal time is the same as the one for a controllable two level system.
We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing the energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. In particular, we prove the 'normality' of the given problem and the 'regularity' of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
We study the manipulation of two-level quantum systems. This research is motivated by the design of quantum mechanical logic gates which perform prescribed logic operations on a two-level quantum system, a quantum bit. We consider the problem of driving the evolution operator to a desired state, while minimizing an energy-type cost. Mathematically, this problem translates into an optimal control problem for systems varying on the Lie group of special unitary matrices of dimension two, with cost that is quadratic in the control. We develop a comprehensive theory of optimal control for two-level quantum systems. This includes, in particular, a classification of normal and abnormal extremals and a proof of regularity of the optimal control functions. The impact of the results of the paper on nuclear magnetic resonance experiments and quantum computation is discussed.
In this paper, we demonstrate an approach to quantum robust control based on the tools of geometric optimal control. The central objects of interest are the sensitivity functions defined as the coefficients in the Taylor expansion of the trajectory with respect to the (unknown, small) parameters which describe the deviation of the actual model from nominal one. In terms of these quantities, we formalize an optimal control problem where one searches for the optimal nominal trajectory which minimizes the size of the sensitivity while taking into account other aspects of the control design such as the energy of the control field. We consider in detail the case of a single qubit with a dephasing Hamiltonian term, and the optimal control problem of obtaining a state transfer by minimizing the weighted sum of the energy of the controlling field and the first order sensitivity. At the limit of a very large weight on the sensitivity, we obtain the optimal control which zeros the sensitivity and minimizes the control field energy. This problem has a rich mathematical structure which enables its solution in terms of elliptic integrals. For this problem, we obtain an explicit solution which is particularly simple and also smooth, avoiding discontinuities which are present in other approaches. We extend the results to the robust control of two quantum bits minimizing cross-talk contamination, as we show that such a problem decouples in two independent one qubit problems.