Yes, you can define a universal quantum Turing machine, which is a single quantum Turing machine U able to simulate any other quantum Turing machine M (at least

Yes, you can define a universal quantum Turing machine, which is a single quantum Turing machine U able to simulate any other quantum Turing machine M (at least, to arbitrary precision) given a coded description of M on its tape. This is one of the main observations David Deutsch made in his famous paper from 1985.