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We know that $BPP \subseteq BQP$ but we have no proof $BPP \subset BQP$ (Though we have the proof that BQP $!=$ BPP with an oracle)

Since Simon's problem (as factoring) it's easily solvable by a quantum computer, and in exponential time complexity solvable by a classical computer, that's a hint of the separation between BQP and BPP and therefore this can be a pure NP problem. Am I right?

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  • $\begingroup$ What research and self-study have you done? There are lots of sources that talk about this topic. $\endgroup$ – D.W. Apr 21 '14 at 14:49
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Simon's problem is not a pure NP problem, for two reasons:

  • It is an oracle problem. We are given an oracle for some function $f$. That's not something that you can do within the definition of a NP problem.

  • It is a promise problem. We are given the promise that $f$ will satisfy a particular property (it is two-to-one, and has a particular structure). That too is not something you can do within the definition of a NP problem.

So Simon's problem is not a problem in the formal complexity class NP; it's just something different. For the same reasons, it's not NP-intermediate, either.

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