Realization of quantum hadamard gate based on Lyapunov method

This paper studies the realization of the quantum Hadamard gate of spin with a single electron. First the canonical form of the unitary rotation gate which is feasibly applied in plane control fields is given. Because the Hadamard gate can't be realized directly by applying the plane control fields, so the matrix operator of Hadamard gate is decomposed into two matrices which are feasible in canonical forms. These two canonical matrices correspond to the rotate-operation and reflect-operation of the vector coordinates on the Bloch sphere, respectively. Then the Hadamard gate can be realized in two steps by choosing appropriate control fields, in such a way the realization of Hadamard gate becomes to design external tailored control fields. Finally, the control fields are designed based on Lyapunov method. Numerical simulation experiments and results analysis are given.