Jarlskog's Parametrization of Unitary Matrices and Qudit Theory

Abstract

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.