  Universal quantum entangling gates are a crucial building block in the
large-scale quantum computation and quantum communication, and it is an
important task to find simple ways to implement them. Here an effective quantum
circuit for the implementation of a controlled-NOT (CNOT) gate is constructed
by introducing a non-computational quantum state in the auxiliary space.
Furthermore, the method is extended to the construction of a general
n-control-qubit Toffoli gate with (2n-1) qubit-qudit gates and (2n-2)
single-qudit gates. Based on the presented quantum circuits, the polarization
CNOT and Toffoli gates with linear optics are designed by operating on the
spatial-mode degree of freedom of photons. The proposed optical schemes can be
achieved with a higher success probability and no extra auxiliary photons are
needed.
