Many futurists estimate that AI will reach and surpass human levels between 2025-2035. All computing will be quantum at this time period if current trends hold. Therefore, I would hypothesize that something a the quantum scale of computing could cause consciousness. However, it is also possible that AI never becomes conscious, or it may be possible that consciousness is an emergent property of complexity. I don't think anyone has the definitive answer at the moment.
Well, a few things. First of all the AI community's not great at these estimations (third paragraph down). Secondly, those predictions aren't founded primarily on the idea of quantum computing (which will change everything, obviously) but on the basis of hitting or coming close to the Singularity. This is based on making a machine (or software) which can create a better machine (or software).
Definitely no one has a definitive answer about what makes consciousness, and similarly, no one has a definitive goal for AI. Because no one knows what they're aiming at, I tend to be pretty skeptical about predictions of how/when they're going to get there. And even still, equating quantum computing and the brain might be a huge mistake- there is definitely no evidence that our brains use quantum computing.