We propose a Lyapunov control design to achieve specific (or a family of) unitary time-evolution operators, i.e., quantum gates in the Schr\"{o}dinger picture by tracking control. Two examples are presented. In the first, we illustrate how to realize the Hadamard gate in a single-qubit system, while in the second, the controlled-NOT (CNOT) gate is implemented in two-qubit systems with the Ising and Heisenberg interactions. Furthermore, we demonstrate that the control can drive the time-evolution operator into the local equivalence class of the CNOT gate and the operator keeps in this class forever with the existence of Ising coupling.
@article{arxiv.1401.3195,
title = {Realization of quantum gates by Lyapunov control},
author = {S. C. Hou and L. C. Wang and X. X. Yi},
journal= {arXiv preprint arXiv:1401.3195},
year = {2014}
}