I would also like to add that another thing that is most likely going to be a problem for quantum computers and that is how fast they can repeat the process. From Graphics cards to Asics the system is designed to do one thing and that is to run one type of calculations as fast as possible. Processors and Quantum computers might be scary fast but (this is just speculations on the quantum computer part) they will be very slow in comparison with running the same calculation over and over again.
Here's the analogy that I'm familiar with. A CPU runs like a bunch of CEOs in the boardroom. They are a few highly intelligent employees of a corporation who exhibit cognitive flexibility, and supreme decision-making skills. Although CPUs serve a vital role in computation, they are ineffectual miners. GPUs are like factory workers who routinely perform the same monotonous operation. GPUs are extremely well-suited to perform mining tasks (e.g., proofs of work). FGPAs compound the benefit of GPUs by sheer numerosity. ASICs are substantially better than CPU for the same reason but they were specifically designed perform the same mundane task over and over.
Your comment above indicates that you'd lump quantum processors in the same group as classical CPUs. What support is there for this contention?
One of the most well-known properties of quantum computers is their ability to concurrently maintain multiple states due to quantum superposition. A qubit can take on the classical values of 0 and 1, but it can also take a third value that conveys a sense of indeterminacy -- both/neither 0 and 1 simultaneously. The way I see it, the speed-up comes in the way the data are processed as a result of prepared inputs. Thus, if the inputs to the hashing algorithm were sufficiently prepared to take advantage of the quantum speed-up, then we'd have a system that could hash exponentially faster than a classical CPU.
A Quantum computer is very specialized. It only "works" for very specific types of problems and I do not believe bitmining is included in that list. It also has to compete with the cost efficiency of ASIC's. Basically it all depends on how the next generations of quantum computers will work. I cant say with certainty that they wont be able to bit mine but I am 80% sure they essentially cant and 99% sure that if they could they would cost more to run then the bitcoins you would earn.