  We study the critical behavior of the random q-state Potts quantum chain by
density matrix renormalization techniques. Critical exponents are calculated by
scaling analysis of finite lattice data of short chains ($L \leq 16$) averaging
over all possible realizations of disorder configurations chosen according to a
binary distribution. Our numerical results show that the critical properties of
the model are independent of q in agreement with a renormalization group
analysis of Senthil and Majumdar (Phys. Rev. Lett.{\bf 76}, 3001 (1996)). We
show how an accurate analysis of moments of the distribution of magnetizations
allows a precise determination of critical exponents, circumventing some
problems related to binary disorder. Multiscaling properties of the model and
dynamical correlation functions are also investigated.
