  In the paper (math-ph/0504049) Jarlskog gave an interesting simple
parametrization to unitary matrices, which was essentially the canonical
coordinate of the second kind in the Lie group theory (math-ph/0505047).
  In this paper we apply the method to a quantum computation based on
multi-level system (qudit theory). Namely, by considering that the
parametrization gives a complete set of modules in qudit theory, we construct
the generalized Pauli matrices which play a central role in the theory and also
make a comment on the exchange gate of two-qudit systems.
  Moreover we give an explicit construction to the generalized Walsh-Hadamard
matrix in the case of n=3, 4 and 5. For the case of n=5 its calculation is
relatively complicated. In general, a calculation to construct it tends to
become more and more complicated as n becomes large.
  To perform a quantum computation the generalized Walsh-Hadamard matrix must
be constructed in a quick and clean manner. From our construction it may be
possible to say that a qudit theory with $n\geq 5$ is not realistic.
  This paper is an introduction towards Quantum Engineering.
