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Electronic and Mechanical Engineering and Information Technology

Optimal control of two-level quantum systems based on the geometrical parametrization

Jiahua Wei, Fengxing Zou, Pengfei Liu

Published Aug 1, 2011• 0 citations
Publisher

Abstract

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.

DOI
10.1109/emeit.2011.6023351

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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.
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.
In this paper, we explore how to constructively manipulate qubits by rotating Bloch spheres. It is revealed that three-rotation and one-rotation Hamiltonian controls can be constructed to steer qubits when two tunable Hamiltonian controls are available. It is demonstrated in this research that local-wave function controls such as Bang-Bang, triangle-function and quadratic function controls can be utilized to manipulate quantum states on the Bloch sphere. A new kind of time-energy performance index is proposed to trade-off time and energy resource cost, in which control magnitudes are optimized in terms of this kind of performance. It is further exemplified that this idea can be generalized to manipulate encoded qubits on the controllable subspace.
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→∞.
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 various physical implementations of quantum information processing, qubits are realized in a Lambda type system configuration as two stable lower energy levels coupled indirectly via an unstable higher energy level, that is, in comparison, a lot more susceptible to decoherence. We consider the quantum control problem of optimal state transfer between two isospectral density matrices, over an arbitrary finite time horizon, for the quantum Lambda system. The cost considered is a compromise between the energy of the control field and the average occupancy in the highest energy level. We apply a geometric approach that combines the use of the Pontryagin Maximum Principle, a symmetry reduction technique to reduce the number of parameters in the resulting optimization problem, and several auxiliary techniques to bound the parameter space in the search for the optimal solution. We prove several properties of the optimal control and trajectories for this problem, including their normality and smoothness. We obtain a system of differential equations that must be satisfied by the optimal pair of control and trajectory we treat in detail, with numerical simulations, and solve a case study involving a Hadamard-like transformation. Our techniques can be adapted to other contexts and promise to push to a more consequential level, the application of geometric control in quantum systems.
In this paper we have synthesized optimal control of multi-level quantum system by representing unitary operator in terms of the projection operators of the Hamiltonian of the system. We have solved the steering problem for multi-level quantum systems minimizing energy cost functional of the control. The optimal control problem of the time evolution of quantum spin of Pauli two-level system subjected to an external field with the minimum energy function is also illustrated and formulated in terms of the quantum spin up and spin down states of the Pauli two-level system.