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arXiv:1010.2805

Optimal bounded trigonometry control of N-level quantum systems based on the geometrical parametrization

Ming Zhang, Weiwei Zhou, Hong‐Yi Dai, Zairong Xi, S. G. Schirmer

Published Oct 14, 2010• 0 citations
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Abstract

Based on a parametrization of pure quantum states we explicitly construct a sequence of (at most) $4N-5$ local time-continuous waveform controls to achieve a specified state transition for $N$-level quantum systems when sufficient controls of the Hamiltonian are available. The control magnitudes are further optimized in terms of a time-energy performance, which is a generalization of the time performance index. Trajectory-constrained optimal local time-continuous waveform controls, including both local sine-waveforms and $n^{\rm th}$-order-polynomial waveform controls are obtained in terms of time-energy performance. It is demonstrated that constrained optimal local $n^{\rm th}$-order-polynomial waveform controls approach constrained optimal bang-bang controls when $n\rightarrow\infty$.

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Optimal bounded local sine-waveform controls for state transfer in NNN-level quantum systems
Based on a parametrization of pure quantum states we explicitly construct a sequence of (at most) 4N−54N-54N−5 local time-continuous waveform controls to achieve a specified state transition for NNN-level quantum systems when sufficient controls of the Hamiltonian are available. The control magnitudes are further optimized in terms of a time-energy performance, which is a generalization of the time performance index. Trajectory-constrained optimal local time-continuous waveform controls, including both local sine-waveforms and nthn^{\rm th}nth-order-polynomial waveform controls are obtained in terms of time-energy performance. It is demonstrated that constrained optimal local nthn^{\rm th}nth-order-polynomial waveform controls approach constrained optimal bang-bang controls when n→∞n\rightarrow\inftyn→∞.
This paper presents an approach based on constructing control Hamiltonians to steer qubit states. When three tunable control Hamiltonians are available, we can adjust the controls to transform two-level quantum systems from an arbitrary initial pure state to another arbitrary target pure state along the unique geodesic curve on Block sphere. In terms of a kind of time-energy performance J = ∫0tf[λ + (1 - λ)E(u(t))]dt, we discuss how to design control magnitude to minimize this performance. An example indicates the feasibility and efficiency of this approach on optimal control of two-level quantum systems.
Abstract. We present a constructive control scheme for solving quantum state engineering problems based on a parametrization of the state vector in terms of complex hyperspherical coordinates. Unlike many control schemes based on factorization of unitary operators, the scheme gives explicit expressions for all generalized Euler angles in terms of the hyperspherical coordinates of the initial and final states. The factorization, when applicable, has added benefits that phase rotations can be combined and performed concurrently. The control procedure can be realized using simple bang-bang or square-wave-function controls. Optimal time-energy control is considered to find the optimal control amplitude. The extension of the scheme to implement arbitrary unitary operators is also discussed.
The time optimal problem for a two level quantum system is studied. We compare two different control strategies of bang-bang control and the geometric control, respectively, especially in the case of minimizing the time of steering the state from North Pole to South Pole on the Bloch sphere with bounded control. The time performances are compared for different parameters by the individual numerical simulation experiments, and the experimental results are analyzed. The results show that the geometric control spends less time than the bang-bang control does.
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.
Recently there has been a lot of interest in the potential applications of performing computations on systems whose governing physical laws are quantum, rather than classical in nature. These quantum computers would have the ability to perform some calculations which would not be feasible for their classical counterparts. To date, however, a quantum computer large enough to perform useful calculations has yet to be built. Before this can be accomplished, a method must be developed to control the underlying quantum systems. This is a problem which can naturally be formulated in the language of control theory. This report outlines the basic control-theoretic approach to time-optimally controlling quantum systems evolving under the dynamics of the Schr\\"{o}dinger operator equation. It is found that under the assumption of non-singularity, the controls which produce time-optimal trajectories are bang-bang. With this in mind, a switching time algorithm is implemented to find optimal bang-bang controls.