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First, I would like to mention the so-called quantum computer is not one of these. It is based on crazy-carpet patches to quantum physics theory that do not make sense and haven't been experimentally verified; besides, beware of recent results announced, as for now no concrete evidence of this absurdity has been shown, and no working model has been devised, let alone built. If you did not hear of it, then do not even bother.
Some fields are much more credible. One of these is simply parallel computing. It deals with computing many things at the very same time, so with way slower but more numerous processors we can beat up a single ultra-fast processor by an exponential factor. Even more interesting is the possibility that, with interconnected computers on a network, processing time could be distributed non-locally; in other words, someday the whole Internet could become some kind of huge computer with millions of processors.
Another alternative does not even deal with electronics. It has been shown recently what DNA could reproduce every basic operation of a computer, and therefore that computers could be based on DNA, even on hybrid biologic-electronic devices, working with very low energy consumption rates, and of course allowed to evolve. I let your imagination do the rest of the work.
Another work based on evolution rather deals with algorithms for now, but could in the near future become fully operational technologies. These are neural networks, intended to imitate the human brain, and genetic programming, imitating genetic transformations. Both focuses on evolution and inaccurate processes, which are of obvious uses in many fields such as artificial intelligence or approximation of otherwise very time-consuming algorithms. Unfortunately, although the results are somewhat astonishing, the technology did not prove to be widely practical...yet.
So much for the big calculator we use to know. In fact, the futuristic computer could do much more than just computing...and maybe even evolve until it has nothing more in common with a computer. The creative process is most elementary regarding those issues. Remember the fifties, when everyone would have sworn there would be robots in every home in year 2000; the technology shifted from a pure machine to something else entirely that has nothing to do with its mechanical counterpart. The future probably reserves us the same kind of surprises.