JARLSKOG'S PARAMETRIZATION OF UNITARY MATRICES AND QUDIT THEORY

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 multilevel 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 ≥ 5 is not realistic. This paper is an introduction toward Quantum Engineering.