The Second Amendment does not merely protect the collective right of the people to bear arms. It protects the individual rights of every person (
see this post for an excellent explanation). Once it is established that the Second Amendment protects individual rights, this question may be answered very easily. A state ban on the possession of an assault weapon is just as unconstitutional as a ban on the possession of any other arms.
It is often argued that the Second Amendment only prohibits the complete denial of the right to bear arms. This line of reasoning, however, is inappropriate. The Second Amendment does not prohibit merely the complete denial of this right, but also any "infringe[ment]."
The Supreme Court of Kentucky interpreted a similar provision in the state constitution in 1822. It ruled in
Bliss v. Commonwealth, "to be in conflict with the constitution, it is not essential that the act should contain a prohibition against bearing arms in every possible form--it is the
right to bear arms ... that is secured by the constitution, and whatever restrains the full and complete exercise of that right, though not an entire destruction of it, is forbidden."
Not everything, of course, is an "arm" within the meaning of the Second Amendment. But a semi-automatic firearm most certainly is, and the right to bear it may not be infringed by any state.