access icon free Optimal control of single spin-1/2 quantum systems

© The Institution of Engineering and Technology
Received 24/04/2013, Accepted 24/09/2013, Revised 31/07/2013, Published

The purpose of this study is to explore the optimal control problems for a class of single spin-1/2 quantum ensembles. The system in question evolves on a manifold in ℛ3 and is modelled as a bilinear control form whose states are represented as coherence vectors. An associated matrix Lie group system with state space SO(3) is introduced in order to facilitate solving the given problem. The controllability as well as the reachable set of the system is first analysed in detail. Then, the maximum principle is applied to the optimal control for system evolving on the Lie group of special orthogonal matrices of dimension 3, with cost that is quadratic in the control input. As an illustrative example, the authors apply their result to perform a reversible logic quantum operation NOT on single spin-1/2 system. Explicit expressions for the optimal control are given which are linked to the initial state of the system.

Inspec keywords: matrix algebra; bilinear systems; Lie groups; reachability analysis; vectors; discrete systems; maximum principle; state-space methods

Other keywords: NOT logic operation; single spin-1/2 quantum systems; bilinear control; state space; associated matrix Lie group system; controllability; coherence vectors; reachable set; optimal control problem; maximum principle; dimension 3 orthogonal matrices; reversible logic quantum operation

Subjects: Discrete control systems; Control system analysis and synthesis methods; Combinatorial mathematics; Algebra; Nonlinear control systems; Optimal control

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