  We investigate the extended quasi-particle states in the mixed state of
d-wave superconductors on the basis of the Bogoliubov-de Gennes equation. We
prove that the quasi-particle eigen-states can be classified in terms of new
topological quantum numbers which are related to the topological nature of the
non-trivial phases of the quasi-particles. Numerical results for the
quasi-particle eigen-states reveal the crossover behavior from gapless to
gapped states as the flux density $B$ increases. In the strong field region
quantum oscillations appear in the excitation energy of the quasi-particles.
