  It is known that every two-qubit unitary operation has Schmidt rank one, two
or four, and the construction of three-qubit unitary gates in terms of Schmidt
rank remains an open problem. We explicitly construct the gates of Schmidt rank
from one to seven. It turns out that the three-qubit Toffoli and Fredkin gate
respectively have Schmidt rank two and four. As an application, we implement
the gates using quantum circuits of CNOT gates and local Hadamard and flip
gates. In particular, the collective use of three CNOT gates can generate a
three-qubit unitary gate of Schmidt rank seven in terms of the known Strassen
tensor from multiplicative complexity. Our results imply the connection between
the number of CNOT gates for implementing multiqubit gates and their Schmidt
rank.
