Realization of quantum gates by Lyapunov control

• A Lyapunov control design is proposed to prepare quantum gates.
• Any one-qubit gates can be implemented.
• The CNOT gate is implemented with different Hamiltonians.
• Local equivalence class of the CNOT gate is achieved.

We propose a Lyapunov control design to achieve specific (or a family of) unitary time-evolution operators, i.e., quantum gates in the Schrödinger 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.