  Considering quantum random walks, we construct discrete-time approximations
of the eigenvalues processes of minors of Hermitian Brownian motion. It has
been recently proved by Adler, Nordenstam and van Moerbeke that the process of
eigenvalues of two consecutive minors of an Hermitian Brownian motion is a
Markov process, whereas if one considers more than two consecutive minors, the
Markov property fails. We show that there are analog results in the
noncommutative counterpart and establish the Markov property of eigenvalues of
some particular submatrices of Hermitian Brownian motion.
