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Is it proven that quantum computation is no better at solving NP complete problems than classical computation or it's just believed?

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It is suspected that NP-complete problems cannot be solved in quantum polynomial time (i.e., that they are not in BQP), but this hasn't been proved. We don't expect a proof in the near future, since this would imply that P is different from NP.

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    $\begingroup$ What about the other way around. If NP-complete is proved to be in BQP, does that say anything about P vs NP? $\endgroup$ – kptlronyttcna Jan 24 '17 at 7:39
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    $\begingroup$ Nothing was known in 2007, though that was quite some time ago. $\endgroup$ – Yuval Filmus Jan 24 '17 at 8:01
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    $\begingroup$ @kptlronyttcna I think it would not say anything on P vs NP as P vs BQP is also unestablished yet. $\endgroup$ – P.Péter Jan 24 '17 at 14:01

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