The core idea of quantum computing is that the particle dynamics of our universe are so inefficiently implemented on the substrate universe that we can harvest some of the substrate computations to escape the computational class of mechanical computation.
In computer science, effective means that a result is reachable, and efficient means that you can get in practice (typically, it means that an algorithm runs in polynomial time).
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Right, but I don't see what privileges polynomial time other than what we're used to at the classical scale. Besides, IIRC, the BQP complexity class is cross cutting and not a simple subset/member in the complexity hierarchy. That's suggestive to me of something rich.
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- If it is in P, we can construct a P time mapping to whatever universal computer we have at our disposal. - Not really. It just means that some problems in BQP can be solved in P time and others cannot. (Probably. There is no proof outside of the NSA whether BQP is not in P.)
End of conversation
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