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Ah, I know it's too late since you already signed off, but on the off chance that you see this, I thought I should clarify. I was referring more to the generalizability of the physically built system and not the algorithm. For example, does a different physical system need to be built to do Shor's Algorithm vs Grover's Algorithm, or can one physical system do both? And then to generalize that further, can a single physical system be built that can implement arbitrary quantum algorithms? Thanks.
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There's no single-sentence answer to "which problems have a fast quantum algorithm?," in exactly the same way as there's no single-sentence answer to "which problems have a fast CLASSICAL algorithm?" The answer is a course or a textbook, not a sentence!
Having said that, the problem solved by Grover's algorithm is actually exceedingly general: it's simply searching through a list of n items for a desired item. And it can be generalized even further than that. The trouble with Grover's algorithm is not at all its generality; rather, it's that the speedup is "merely" quadratic.