Complexity inversely propotional to $n$
I'm curious if anyone's come up with a problem or method as n => infinity t => 0. Are there any sort of cases found in quantum computing?
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As the complexity of an algorithm is a measure of the number of operations (in a sense to be defined in each context) needed to do some computation in function of the size of some input, sub-constant complexity does not make any sense. With your exemple, $O(\frac1n)$, it means that for a sufficiently large input, the algorithm does strictly less than one operation, which in terms of Turing machines means that the initial state is accepting, which means that the corresponding Turing machine does not output anything.
Edit: I had not seen the quantum computing reference, so my answer might not be total, although I doubt it makes sense even in that context.
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