Construction of General Unitary Transformation for Conditional Quantum Dynamics and Realization of Quantum Controlled Gates

By factorizing a general one-bit transformation matrix for one-bit gates in quantum computer, a general unitary transformation is constructed, which can serve as (controlled)^m gate for the conditional quantum dynamics. When the quantized single-mode electromagnetic field containing n or no photons acts as the controlled bit, the quantum controlled-NOT gates and square root of the controlled-NOT gates in cavity-quantum electrodynamics are realized.