No it doesn't. You know that the result of the computation for an individual qubit will be either 0 or 1 (otherwise it would be useless -- measuring only gives you one bit of information), so you construct the system such that after the computation is done each qubit will be aligned with the |+z> or whatever axis. The key point is that you have to be clever about how you construct the system for a given QC algorithm, not that you cannot do arbitrary computations using the system.